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A.J. Bishop et al.(eds.), International Handbook of Mathematics Education, 1996, Kluwer Academic Publishers, 1097-1121


Chapter 29: Didactics of Mathematics and the Professional Knowledge of Teachers


PAOLO BOERO, CARLO DAPUETO, AND LAURA PARENTI

Department of Mathematics, University of Genoa, Italy


ABSTRACT


This chapter will deal with the problem of the relationships between didactics of mathematics as a rapidly developing field of investigation and the professional knowledge of mathematics teachers as an individual and social construction. We will focus on teachers' pre-service and in-service education as the situations where systematic contacts may be established between research in didactics of mathematics and the construction of teachers' professional knowledge.

    We will discuss present difficulties in establishing a productive relationship between teachers' education and different strands of research in didactics of mathematics, and different perspectives about how to establish such a relationship.

 

1.   INTRODUCTION

2.   THREE EXTREME ORIENTATIONS IN MATHEMATICS TEACHER EDUCATION

2.1   The teacher must become more and more competent in mathematics: 'He who knows mathematics, knows how to teach it'.

2.2   The teacher must develop his/her professional competence like an artisan (if possible, an artist)

2.3   The teacher's professional competence must be grounded in different scientific domains (mathematics, sciences of education, didactics of mathematics)

3.   DIFFERENT MODELS OF RESEARCH IN DIDACTICS OF MATHEMATICS AND THEIR IMPLICATIONS FOR MATHEMATICS TEACHER EDUCATION: WHY IS IT SO HARD TO CREATE A PRODUCTIVE DIALOGUE BETWEEN RESEARCH IN DIDACTICS OF MATHEMATICS AND MATHEMATICS TEACHER EDUCATION? HOW TO IMPROVE THE SITUATION?

3.1   Our general orientation

3.2   The present situation

4.   DIFFERENT METHODOLOGICAL PERSPECTIVES CONCERNING TOOLS AND RESULTS OF RESEARCH IN DIDACTICS OF MATHEMATICS IN MATHEMATICS TEACHER EDUCATION

4.1   Involvement of future teachers and/or in-service teachers in research in didactics of mathematics, which frequently implies a model of teacher as researcher (although not in the sense of a full time job)

4.2   Introduction of research topics in didactics of mathematics in more or less traditional 'courses'

4.3   By carrying out problem-solving activities prospective teachers directly experience what constructive learning means and what kind of difficulties it may involve

4.4   Discussing professional problems personally met in the class (during pre-service apprenticeship, or normal teaching experience)

4.5   Discussing selected materials, videotapes, quantitative results, etc. resulting from research work or from other teachers' class experiences.

4.6   As a conclusion concerning methodologies

5.  CONCLUSION

6.  REFERENCES


1.   INTRODUCTION


In less than one century, mathematics changed from a set of elementary tools needed in everyday life and a specialised domain of investigation, to a pervasive component of today's culture: mathematics is deeply embedded in technology and in many aspects of today's manner of viewing natural and social phenomena. Mathematics became a general language to represent reality and a powerful, flexible simulation tool. This increasing need for mathematics education developed in parallel with another, rather contrasting orientation: in many countries mathematics took the place of Latin (or Greek) as the subject matter responsible for school orientation and selection.

    After the Second World War mathematics teacher education became more and more complex for other, different reasons: the increasing number of students for each age group made the profession of teaching more difficult (if the teacher wants to or is required to ensure efficiency in his activity)-especially in the case of mathematics because of its specific difficulties. In the field of mathematics, like in other fields, the impact of new technologies has changed some priorities within the educational aims (making the previous curriculum partly obsolete); it has also brought on the need for new competencies on the teacher's part, in order to take advantage of the new educational opportunities offered by new technologies and better understand their cultural and social impact (see: Ciosek, 1990; Keitel, 1986; NCTM, 1991; Boero, 1992, 1993). The speed of changes (in mathematics, and in the school system) in itself provoked a crisis in the old manner of conceiving the profession of the teacher as an art, personally developed through apprenticeship in the school environment and based on a good knowledge of mathematics. Another reason for crisis depends on the fact that in many countries some teachers become teachers for reasons which are very far from a genuine vocation to teach mathematics. Their apprenticeship as mathematics teachers may be strongly influenced by this lack of specific motivation to teach mathematics.

    A crucial point of our chapter will concern the need for research in didactics of mathematics in relationship with today's situation of mathematics education and the profession of mathematics teaching, and the related need that pre-service and in-service teacher education introduce teachers to and involve them in research methods and results.

    Present research in didactics of mathematics (a rapidly developing field with different trends and schools) may offer some tools in order to increase the effectiveness of pre-service and in-service mathematics teacher education and provide support to the profession (see: Arzarello & Bartolini, 1994; Boero & Szendrei, 1994; Cobb, Wood & Yackel, 1990; Cooney, 1994, 1994 b; Krainer, 1994; Wittman, 1991).

    In this chapter we will consider the relationships between didactics of mathematics as a domain of investigation and the professional knowledge of mathematics teachers as an individual construction, socially situated (in the school system and society contexts) and a social construction (performed inside formal or informal groups of teachers).

    We will try to move from our personal experiences and points of view towards a more general perspective, including other points of view. Some examples will be taken from our direct experiences of cooperative work with teachers, concerning research and innovation in the teaching of mathematics, other examples will derive from the Italian situation. The aim of these examples is to provide concrete references for general statements and perspectives; the reader may find similar examples in the reality of other countries (and some references will be provided in this direction).

    The subject matter of this chapter will be organized according to two criteria:

–  a classification of different strands of research in didactics of mathematics, based on the different nature of research results. This criterion was chosen because we think that both the involvement of teachers in research and the exploitation of research results in the school system strongly depend on the characteristics of the different strands of research (especially for what concerns the nature of their results);

–  the idea that the relationships between didactics of mathematics and the professional knowledge of mathematics teachers especially concern pre-service and in-service teacher education. This criterion was chosen in order to show some direct implications of our general analysis about the different types of research results on the teaching profession; indeed, in many countries teacher education is the most important occasion for teachers to encounter research perspectives and results.

The relationships between research in didactics of mathematics and mathematics teacher education may be considered under different points of view, according to the ideas people have of both research in didactics of mathematics and mathematics teacher education. In this chapter we will try to give a general outline of the problems. First of all, we will focus on some current ideas of mathematics teacher education, trying to point out some historical perspectives and present motivations. Then, we will discuss what kind of tools and results which research in didactics of mathematics offers today may be introduced into mathematics teacher education, comparing present needs with actual offers and pointing out some possible directions in order to improve the present situation. Finally, we will try to explain some specific methodological issues, concerning the introduction of results and tools of research in didactics of mathematics into mathematics teacher education, relatively independent of the choice of a peculiar orientation in the field of research in didactics of mathematics.


2.   THREE EXTREME ORIENTATIONS IN MATHEMATICS TEACHER EDUCATION


We will consider different ideas people have of mathematics teacher education. We will describe only 'extreme' positions, although intermediate ones are possible, in order to make different orientations clearer which explicitly or implicitly have contrasting influences on political decisions and their practical implementation (see: Barra & al., 1992; Boero, 1993; Bottino & Furinghetti, 1994,1994b; Furinghetti,1994; Houston, 1990; Hoyles, 1992; Ponte & al., 1994).


2.1   The teacher must become more and more competent in mathematics: 'He who knows mathematics, knows how to teach it'.


This traditional idea in its rough version is not popular amongst mathematics educators, but it is still very common amongst mathematicians and mathematics teachers (especially in high schools), with possible misunderstandings about mathematics ('school mathematics' or 'mathematician mathematics'?). 'I've think that the reasons for the widespread permanence of this idea might he an interesting subject for research in didactics of mathematics. Does it de pend on the need to protect some specific 'competence' and related 'criteria of quality' pertaining to the domain of mathematics; or on the difficulties of mathematicians and mathematics teachers in evaluating and taking advantage of contributions deriving from scientific domains which are very far from mathematics (like sociology of education, psychology etc.); or on the presumption of being able to cover all the needs concerning the 'transmission' of mathematical knowledge with a sufficiently deep mastery of the subject field (mathematics) and some professional expertise obtained through ongoing professional experience?

    As to the exclusive focus on 'knowing mathematics' as a requisite to become a teacher, we think that is it important to consider a possible, more sophisticated (and not so popular) version of this conception: 'knowing mathematics' might include knowledge of history of mathematics, epistemology of mathematics, philosophy of mathematics. Different perspectives in these fields are given by books and articles such as: Davis & Hersh (1980); Kline (1980); Lakatos (1967); Steen (1978); Goodman (1980); Mac Lane (1981); Swart (1980); Dapueto & Ferrari (1983). This broad competence might allow teachers to understand both the relationships between concept construction, formalization and theoretical framing in the domain of mathematics, and the relationships between mathematics and other cultural domains (physics, philosophy, etc.). According to this sophisticated conception, some strands of research in didactics of mathematics might have something to say about teacher education (see: 'epistemological obstacles', in Brousseau, 1983 and Sierpinska, 1985, 1987; 'didactical transposition', in Chevallard, 1991, etc.). In some countries (e.g.: France, Germany, Italy) there are still traditions or new experiences developed, in which 'knowing mathematics' in a broad, cultural sense is considered as a crucial aspect of mathematics teacher education, and some researchers in didactics of mathematics are involved in it. Unfortunately, in those cases there is a dramatic contradiction between ordinary, technical education in mathematics (ensured through traditional, non-interactive technical lectures and exercises concerning isolated and specialized fields of today's mathematics) and some courses or seminars where prospective teachers (or in-service teachers) get to know different historical and epistemological perspectives. In short, this contradiction results in a 'cultural varnishing', with no practical, deep influence on professional choices and classroom activities.


2.2   The teacher must develop his/her professional competence like an artisan (if possible, an artist)


We will use the words 'artisan', 'artist' to evoke the idea of a teacher who is able to face professional problems in a flexible way ('artisan') or to create substantial innovations ('artist').

    This conception, too, is very popular amongst mathematics teachers and mathematicians. Its implications for mathematics teacher education may include the preceding conception, with a special emphasis on apprenticeship, usually at the end of the mathematics preparation or while taking up teaching: 'he who knows mathematics, knows how to teach it' is replaced by 'a good mathematics teacher must master mathematics and be acquainted with the art of teaching'. This conception can be better understood under a historical perspective, and recognized as one of the sources of present research in didactics of mathematics (although many researchers in mathematics education today refuse it).

    Up to the 70's, this conception was shared by many 'mathematics educators'. At that time, 'innovation' was considered the most valuable and useful outcome of 'investigation', so 'artists' (those who produced new brilliant ideas about teaching specific subject matters, or new methodologies) had a place in mathematics teacher education (especially in the in-service mathematics teacher education). Besides 'innovations', 'artists' frequently produced general ideas about teaching of mathematics as an 'art', in the form of 'principles', or 'general orientations', or 'comments about experiments'. Such ideas usually involved a wide range of considerations with different levels of deepening, depending on competence, background and traditions from rather deep epistemological or historical analysis about mathematics in general and specific mathematical subjects, to rather naive (and not well framed) considerations about learning of mathematics, assessment, sociology of mathematics education. In many countries, research in didactics of mathematics (as we nowadays know it) has developed sometimes (Hungary is an example) in continuity with, sometimes (France) in opposition to these 'general ideas' about teaching of mathematics as an 'art'.

    Despite the importance attributed in the field of sciences of education to 'apprenticeship', today few researchers in didactics of mathematics agree with the idea of teaching as an 'art' learned through a suitable apprenticeship, an idea which excludes (or reduces) the importance of scientific knowledge about didactics of mathematics in pre-service and in-service teacher education. But sometimes mathematics teachers recognised as 'artists' seem to have a very strong influence on teaching of mathematics (...stronger than the community of researchers in M.E.). Is this true? Are those influences beneficial to mathematics education? Another interesting research topic for research in didactics of mathematics! We should distinguish the different levels at which the influence of 'artists' works today. If we take into consideration the environment of teachers involved in innovation, comparison of ideas, etc. (very few teachers who join in associations of mathematics teachers, take part in regional or national or, some of them, international meetings, etc.), the influence of 'artists' is important and beneficial. But today many mathematics teachers in the world do not take part in 'movements', do not read specialized reviews for mathematics teachers, are not involved (as volunteers) in innovations. In this case, the 'artists' contributions are taken as an alibi not to change anything: 'they do it, but I am not so involved as they are, so well trained as they are, etc., so I cannot do it'. Actually, important changes occurred in the 80's all over the world. With some exceptions (e.g. Hungary and U.K.), in most countries the teaching of mathematics became less and less important socially, and more and more people took it up as a second (or third) choice job. For instance, in Italy now we may estimate that only few young teachers have chosen this profession because they like teaching mathematics (frequently, they like neither teaching, nor mathematics!).

    Another issue is related to 'how to become a good artisan': in the case of mathematics teachers, strong personality requirements seem to be necessary. In other words, it is not sufficient (although it is necessary!) to like mathematics and teaching in order to become an efficient 'artisan' (or, especially, an 'artist'). In our experience of mathematics teacher education (especially, inservice education) we have frequently met motivated people, frustrated because they are not able to manage classroom work as they would like, or to quickly penetrate the student's thinking process in order to interact productively with him/her, or to make real time connections between different subjects, in order to increase the learning opportunities offered to students during a discussion.... More likely, an efficient 'artisan' (especially, an 'artist') is supported by important psychological resources (a strong personality, openmindedness towards introspection, quickness in real time interacting and making cultural connections,...). However, if the model of the good teacher is that of an 'efficient' artisan, the lack of these attributes may result in an increasing de-motivation towards the profession of teacher.

    Can present research in didactics of mathematics offer opportunities for increasing professional performances and satisfaction to all teachers (including those who do not succeed as artisans or artists, even though devoted to their work; and those who choose the profession of teacher as a second or third choice job)? Today this is a good challenge for research in didactics of mathematics, whose outcomes may result in an important motivation for future development of research in didactics of mathematics - or in a big failure of it. In this sense, it is very important to follow what will happen in the future in countries such as France or Canada, where important groups of researchers in didactics of mathematics may reach a large number of teachers in their preservice and in-service vocational education.


2.3   The teacher's professional competence must be grounded in different scientific domains (mathematics, sciences of education, didactics of mathematics)


Teacher education must enable prospective teachers to widen their knowledge connected to the relevant field (mathematics) providing a well-balanced mixture of different subjects related to the different school levels, different sciences of education (from psychology of learning to sociology of education) and mathematics education (as a specific field of professional competence and/or as a specific field of research).

    Since the Second World War, this conception has been more and more extensively represented and realized at political and academic level, particularly after the 60's. Mathematics educators and many teachers (especially at the level of compulsory education) share this conception; but behind their general agreement we can recognize very different orientations, concerning:

–  the institutional environment where knowledge and skills can be developed: pre-service and/or in-service mathematics teacher education? In case of pre-service mathematics teacher education, through 'in series' or 'in parallel' subjects? And what about the academic environment for pre-service mathematics teacher education: the department of mathematics of the 'faculty' (or 'institute') of sciences of education, the department of mathematics of the 'faculty' (or 'institute') of exact sciences, or an ad-hoc academic environment for teacher education? Each of these solutions is experienced (or has been experienced, or is scheduled) in some countries of the world! Sometimes the solution consists in a cooperative, parallel process, including courses of mathematics and mathematics education offered by the department of mathematics and courses of general education, psychology, etc. offered by the 'faculty' (or 'institute') of sciences of education. Sometimes an 'in series' process takes place in different academic environments starting with mathematics education followed by vocational education. Sometimes different solutions are carried out in the same country (Hungary is an example), according to different levels of schools and education;

–  the proportion of the different subjects in pre-service and in-service mathematics teacher education, depending on traditions, school levels, academic environments for mathematics teacher education, academic power of the different scientific groups;

–  the methodology, especially as far as it concerns professional competence. As to this issue, different options may be considered (and different options are concretely practised in the world): from lectures to interactive courses, from individual problem-solving sessions, to working groups reflecting on students' behaviour. We will develop this point in the fourth section. We must point out that unfortunately few experiences concern effective methodologies to teach mathematics to future teachers (Canada, Hungary and U.K. offer some interesting examples). In many other countries, mathematics courses in pre-service mathematics teacher education follow the model of traditional university courses in mathematics (sometimes at a lower level than required to become a mathematician). And this may have deep, negative influences on future professional orientations (very often, university students of today will reproduce tomorrow, as school teachers, the model of their university course!)

–  the content of education in didactics of mathematics. We will develop this point in the following section. Here we remark that there are two extreme positions in the academic environment. Those (mathematicians or mathematics educators, or researchers in the domain of sciences of education) who do not recognize the value of mathematics didactics research results and tools in mathematics teacher education press for including in it only professional techniques (frequently identifying vocational education with apprenticeship). On the contrary, those who recognize the value of mathematics didactics research results and tools in mathematics teacher education tend to emphasize (both for pre-service and in-service mathematics teacher education) those specific results and tools and submit the reflection on apprenticeship to criteria of analysis for research in didactics of mathematics.


3.   DIFFERENT MODELS OF RESEARCH IN DIDACTICS OF MATHEMATICS AND THEIR IMPLICATIONS FOR MATHEMATICS TEACHER EDUCATION: WHY IS IT SO HARD TO CREATE A PRODUCTIVE DIALOGUE BETWEEN RESEARCH IN DIDACTICS OF MATHEMATICS AND MATHEMATICS TEACHER EDUCATION? HOW TO IMPROVE THE SITUATION?


This part of the chapter deals with the contributions that present research in didactics of mathematics may give to mathematics teacher education. Nowadays, research in didactics of mathematics is not a homogeneous reality in the world. There are different traditions and schools, sometimes in the same country, depending on local and general factors concerning the history of both the community of mathematicians and the community of researchers in sciences of education, and the history of mathematics education in that country (for some different perspectives, see: Artigue, 1990; Arzarello & Bartolini,1994; Balacheff,1990; Barra & al., 1992 Boero,1993; Chevallard,1992; Kilpatrick,1992; Steiner, 1984)

    In our opinion, a good way to classify the different kinds of research in didactics of mathematics now existing in the world, establishing a close relationship with their possible implications in mathematics teacher education, is to consider the nature of their possible outcomes; following Boero & Szendrei (1994), we may consider different kinds of results in research in didactics of mathematics:

i)   'innovative patterns' to teach a specific subject (old or new for school mathematics), or to develop some mathematical skills; or, more generally, innovative methodologies, curricula, projects, etc. Results may consist in innovative educational material, 'proposals', or reports about innovations or projects that have been experimented. Frequently, teachers take part in the production of these results as researchers or researchers' partners who perform teaching experiments. An important variable is the dimension of innovation (both in terms of time: a short sequence, an innovative five year curriculum; and in the terms of content: a specific subject, or an integrated system of topics and methodologies). Other variables depend on philosophical grounding and purposes of innovation, theory-practice relationships, etc.;

ii)   'quantitative information' about the results of educational choices related to the teaching of a specific mathematical subject; general methodologies; curricular choices (including comparative and quantitative studies). Or quantitative information about general or specific difficulties regarding learning of mathematics, and their possible correlations with factors influencing the learning process. Information is based on quantitative data, collected and analyzed according to standard or ad hoc statistical methods; the level of statistical treatment may be elementary (only percentages and histograms) or quite sophisticated. Usually, research is performed by researchers who are not school teachers;

iii)   'qualitative information' about the results of some innovations as to methodology or content, or some general or specific difficulties concerning mathematics, etc. In this case, information is based on careful consideration of students' papers, of recorded teacher-student interactions, of recorded group or class discussions, etc.; frequently, these analyses implicitly or explicitly refer to general educational or psychological or didactical theories. Frequently, teachers take part in the production of these results as 'participant observers' (Eisenhart, 1988), sometimes as researchers.

iv)   'theoretical perspectives' regarding the relationship between 'teacher', 'students' and 'mathematical knowledge' in the class; the role of the mathematics teacher in the class; the relationships between school mathematics and mathematicians' mathematics; topics to be taught; the relationships between research results and classroom practice in mathematics education, etc. These results may involve descriptions and classifications of 'phenomena', interpretations of 'phenomena', 'models', historical or epistemological analysis (oriented towards educational aims) of a topic, etc. In most cases, teachers have a marginal role in the production of these results.


3.1   Our general orientation


We agree with Boero & Szendrei (1994) that results of the 'qualitative information' and 'theoretical perspectives' types are important not only in themselves, but also because they allow teachers and researchers to keep other results under control, while results of the 'innovative patterns' and 'quantitative information' types immediately provide teachers with working tools. We may add that all these different results should be provided to teachers during both pre-service and in-service education, in order to help them accomplish their professional choices constructively and critically.


3.2   The present situation


Nowadays, this ideal orientation is very far from reality both as to the relative popularity of different kinds of results amongst teachers, and as to the results of different kinds actually offered by researchers. In particular, teachers tend to neglect some results which might be very useful for them, and researchers offer results which frequently do not fit teachers' needs. We will try to explain this mutual 'incomprehension' and point out some possible directions in order to improve the present situation.


3.2.1 The lack of a common vocabulary and a common background


At present, the results of the 'innovative patterns' and 'quantitative information' types are the most accessible and known for mathematics teachers, especially secondary school teachers, although their application at schools may often result in frequent malfunctions and failures. Several reasons may explain this fact; in our opinion, one of them lies in the teachers' background. Most secondary school mathematics teachers have only a background in mathematics and, possibly, experimental sciences. On the other hand, the research and related results of the 'qualitative information' and 'theoretical perspectives' types need a specific 'human sciences' vocabulary, and are necessarily more overtly grounded on philosophical assumptions.


3.2.2 Failure of promising innovations


Innovations are demanded as products of research in didactics of mathematics, even though their diffusion has not always been successful and in spite of the attack by some mathematics educators on the idea of 'change through innovation' (see the paper by J. Robinson, in Clements & Ellerton, 1989).

    Useful and successful innovations (i. e. results of the 'innovative patterns' type) might be very important to justify (in mathematicians and teachers' eyes) systematic connections between research in didactics of mathematics and mathematics teacher education. Why did many promising 'innovations' result in a failure when they had been so widespread?

    As to this issue, some phenomena should be quoted. We will consider only some examples of innovations concerning specific mathematical subjects. In the past twenty years, mathematics educators have been concerned with the distance between proposals and prototypical innovation, and widespread innovation about subjects like the 'set' approach to natural numbers, or 'ratio, proportion and linear functions'. A comprehensive research perspective is lacking, and few research papers tackling this subject can be offered to teachers during mathematics teacher education in order to help them understand what happened (Brousseau, 1980, 1981, 1986).

    Other phenomena refer to 'popular' and 'not popular' innovations: in Italy, the introduction of substantial topics of elementary probability in the comprehensive school (even if many prototypes are available) meets with many obstacles (Belcastro, Guala & Parenti, 1986); on the other hand, the introduction of elements of analytic geometry in the comprehensive school has taken only few years to succeed. Concerning probability, the situation in U.K. and U.S.A. is quite different! It is not easy to tackle these differences (which might be very interesting to deal with in mathematics teacher education) in a research perspective in didactics of mathematics.

    In general, we must face the problem of the conversion of proposals and prototypes (typical of a research environment) into widespread innovations; it is well known that it is very difficult to reproduce innovations on a large scale without substantial degeneration and loss of effectiveness (cf. Arsac, Balacheff & Mante, 1991; Artigue, 1990).

    As researchers, we think that intercultural differences should be taken into account, because frequently the situation is not the same in different countries; other aspects should also be considered: in-service teacher education; the institutional aspects (official programs, relationships with the preceding and following levels of school, structure and content of final examinations); specific ' didactical transposition' (Chevallard,1991) problems; difficulties in integrating new subjects in the old curriculum and/or changing some parts of this in the perspective of new subjects (Dapueto & C, 1994); the incidence of textbooks; the distance between the community of researchers and the school system (a good integration is realized only in few countries). But there are few research articles and surveys available on these issues!

    Generally, teachers should understand what 'variables' may cause innovations to succeed (or to fail or degenerate) and to be reproduced on a large scale in the field of mathematics education. Research in didactics of mathematics should provide teachers' tutors and teachers with the opportunity of developing and/or exploiting careful and comprehensive descriptions and analysis of:

–  class teaching-learning processes, and (possibly) their interpretations and models;

–  students' long term learning processes, in order to better understand many aspects of the development of their mathematical knowledge, which depends on individual and social factors, cultural influences and emotional constraints, etc.;

–  teachers' conceptions in different countries.

The results of the 'qualitative information' and 'theoretical perspectives' - types are needed, in order to go beyond the present, very limited knowledge of some of these aspects on teachers and teachers' tutors' part. In particular, teachers' conceptions and students' mathematical experiences in everyday life should be carefully investigated because they can deeply affect class teaching-learning of mathematics (see Jaworski, 1994); we would also like to point out that this is the reason why, frequently, 'quantitative' information provided by international comparative studies is biased because of the lack of knowledge about these aspects.


3.2.3 The decreasing involvement of leading research schools in didactics of mathematics in carrying out innovations


If we agree that teachers need the results of the 'innovative patterns' type and that research in didactics of mathematics must produce them, then these results must be spread and compared through international journals and meetings of mathematics educators; but if we actually read outstanding journals and take part in specialized international meetings we find fewer and fewer 'didactical proposals' or reports about innovations.

    For instance, we estimate that from 1970 to 1974 over 60 per cent of the papers published in Educational Studies in Mathematics dealt with the results of 'innovative patterns' type, while from 1987 to 1992 less than 15 per cent of the papers dealt with these. The situation of other outstanding journals is similar. Fewer and fewer results of the 'innovative patterns' type are considered as outstanding 'research outcomes', worth publishing by important journals. And even in many rather specialized international meetings of researchers (like the conferences of the International Group for the Psychology of Mathematics Education) the introduction of 'innovations' takes place especially in the poster session.

    Actually nationally circulated journals continue publishing a good number of papers of that kind, but they are not considered (in most countries) to be 'research' journals. On the other hand, if we consider most of the results of the 'innovative patterns' type presented at congresses or in local journals, we see that they are limited to educational material and/or descriptions of proposed (or experimental) short or long educational sequences, and do not try to go through the related educational, epistemological or psychological problems. The consequence of this fact is that the conditions under which innovations can be reproduced, variables which affect effectiveness, etc., are not known.

    Taking into account all these remarks, a stronger involvement of research in didactics of mathematics in innovation and a proper style of presentation of results of the 'innovative patterns' type is needed for mathematics teacher education; as to this issue, we think that the presentation should be supported and framed by already existing results of the other types (especially of the 'qualitative information' and 'theoretical perspectives' type), in order to make them valuable as research results.

In particular, if the innovation turned out to be a successful experiment, a careful consideration of the conditions which allowed such success (for instance: teachers' motivation, teachers and students' cultural background, school traditions, etc.) should be provided, trying also to point out possible limitations of the 'reproducibility' of the innovation. In an experimental program concerning innovation, phenomena such as 'obsolescence of innovation' (with consequent effects on the teaching-learning process) should be taken into account.

    Here is another example: the epistemological analysis of a mathematical topic, as well as the analysis of the relationships between current and historical points of view in mathematics and in school mathematics concerning it, seem to be necessary in order to make an educational proposal concerning that topic from the cultural point of view: a teacher must therefore be encouraged to take some distance from his/her cultural background; and possible ,epistemological obstacles' inherent in that topic must be anticipated.


3.2.4 Paradigmatic mathematical topics or topics which might be interesting for teachers?


If we agree that research in mathematics education must keep in contact with present mathematics and thus help teachers teach topics which are relevant to modern views about mathematics, or take into account the opportunities offered by new technologies, research should then create connections with these modern views and opportunities. But, if we read articles published by leading international M.E. journals, we will find out that most of them deal with 'paradigmatic' topics and problems (i.e. topics and problems which allow the comparison of new research results and perspectives with previous ones), regardless of the importance of the content or problems in mathematics education (while many traditional and new fields are not covered). And results of the 'innovative patterns' type also focus on only a few directions (regardless of the importance and difficulty of the subject).

    Here is an example about this issue: today 'rational numbers, decimal numbers and approximations of numbers in a calculator or computer environment' form a very important field for everyday choices in mathematics classes which is, however, almost neglected by the main journals. And here is another example: with the exception of spreadsheet, there are few research studies in didactics of mathematics dealing with the educational opportunities offered by professional software in the field of mathematics education and the new educational needs which depend on their diffusion out of the school system.


3.2.5 The results of 'quantitative information'


Some results of the 'quantitative information' type (especially those presenting 'objective', easy to read comparative data) are very popular amongst school teachers and administrators; some mathematicians consider them as the only 'scientific' results in didactics of mathematics. We think that the dissemination of those results must be seriously considered by researchers in didactics of mathematics because they provide teachers as well as parents, school administrators, etc. with information which often seems to be 'objective' and 'scientific'. Taking this aspect into account, the quantitative evaluation of students, teachers, school systems, projects and innovations needs special attention, as it may cause serious damage (for instance, it may orient the teachers' work only towards preparing students to be successful in assessment tests).

    This is one of the main reasons why we think that teacher education must provide teachers with the results of both the 'qualitative information' and 'theoretical perspectives' types: they need them in order to keep the methodologies and the results of the 'quantitative information' type under control. For instance, in our opinion it is crucial that teachers be acquainted with the following problem: is it really possible to keep variables 'constant' in order to create effective control groups? In fact, such a problem deeply affects the comparison of results among different classes, different teachers, different methodologies ..... The nature of tests is another important problem: the debate about different kinds of tests (multiple choice, open questions, etc.) is very important for mathematics education, because each kind of test provides information about specific skills and knowledge. Teachers should become familiar with this problem; indeed, their choices may deeply affect the evaluation of students' performances and class work.

    It would also be important to explain in mathematics teacher education why some 'laboratory' results of the 'quantitative information' type, which may be interesting from a psychological or sociological point of view, may be scarcely relevant to mathematics education and, in any case, must be critically considered. Actually, correlations are often interpreted as cause/effect relationships and some effects of educational choices are confused with epistemological difficulties!

    In short, results of the 'qualitative information' and 'theoretical perspectives' types are useful for teachers in order to make use, in the field of mathematics education, of experimental and statistical methodologies borrowed from sciences of education in a critical way. Unfortunately, research in didac tics of mathematics today provides few understandable, critical materials in this domain, that are suitable to be used in current pre-service and in-service mathematics teacher education


4.   DIFFERENT METHODOLOGICAL PERSPECTIVES CONCERNING TOOLS AND RESULTS OF RESEARCH IN DIDACTICS OF MATHEMATICS IN MATHEMATICS TEACHER EDUCATION


Prospective and in-service mathematics teachers learn about tools and results of research in didactics of mathematics according to different methodologies such as the following:


4.1   Involvement of future teachers and/or in-service teachers in research in didactics of mathematics, which frequently implies a model of teacher as researcher (although not in the sense of a full time job)


As to pre-service teacher education, it takes place in a research environment: students take part in classroom experiments (as observers or as 'participant observers', Eisenhart, 1988), frequently they take part in planning experiments and working out data. Usually students must prepare a thesis or a report on their research experience, sometimes they cooperate in writing reports or articles with experienced researchers. Evidently, students are more oriented towards profession and change when they are directly involved and innovation and innovative experiments take place.

    During the 70's, about one hundred students at Genoa University took their first level mathematics degree discussing a thesis about their one-year voluntary experience in a secondary school classroom. The mathematics teacher and the student were involved in long-term innovations planned in a large, cooperative groups of university researchers and teachers.

    The long-term effects of that experience are generally considered positive by all partners involved (researchers, classroom teachers, prospective teachers). Some of those students became teachers-researchers and are cooperating in the research activities of our group. Similar experiences were performed in other Italian universities.

    Unfortunately, those experiences had no consistent development in the _ 80's and 90's for different reasons which partly depend on the specific Italian situation. Actually, no 'reform' of the pre-service secondary school teacher education was carried out at University level, and the inertia of the university system became stronger and stronger and discouraged volunteer professional experiences in the classroom. Some other reasons seem to be more general and interesting to compare with other realities: when the research work was relatively simple and strictly connected with classroom innovations, it was possible to involve in it mathematics students with no specific background in mathematics education or in general education. Now the situation is completely different: research activities include relatively deep epistemological_ reflections about mathematics, teaching experiments are quite sophisticated, the effective observation and analysis of what happens in the classroom needs a relatively important explicit or implicit background in the field of mathematics education. Students, when starting work for their degree thesis in mathematics education, need a wide and deep specific background. The 144 hours of university courses available today at Genoa University in the field of mathematics education for prospective mathematics teachers are not sufficient to provide students with that preparation.

    As to in-service mathematics teacher education through involvement in research in didactics of mathematics, very interesting experiences are performed in some countries (Australia, Austria, Canada, U.K., etc.). Also the Italian experience seems to be advanced and qualitatively interesting in the world scene, although involving only few teachers (Arzarello & Bartolini Bussi, 1994; Barra & al., 1992; Boero, 1993). Most Italian research groups in didactics of mathematics involve primary and/or secondary school teachers in a long term, deep experience of cooperative work which includes planning of teaching experiments, 'participant observer' experiences in the classroom, and analysis of what happened in the classroom according to different theoretical frameworks. These experiences show very interesting outcomes for inservice mathematics teacher education: teachers overcome the individualistic, 'isolationist' idea of their profession and learn to cooperate with one another. They learn to use both research tools and results to plan, observe and evaluate their classroom work. They learn to take some distance from their classroom experience and to profit from other people's experience.

    Unfortunately, these in-service education experiences involve only very few teachers on a voluntary basis, and their influence on the whole school system is still very weak. On the other hand, sometimes these teachers-researchers have a strong influence in their own school environment because they try to introduce methods of cooperative discussion with their colleagues, elementary research tools useful to interpret classroom phenomena, up-todate views about mathematics and mathematics teaching. Teachers may discover that their profession can functionally include stimulating cultural activities (posing problems, explaining phenomena, etc.) Finally, the ability to make good use of teachers-researchers' skills seems to depend on the school policy (the head of the school plays an important role in it!).


4.2   Introduction of research topics in didactics of mathematics in more or less traditional 'courses'


Nowadays, mathematics educators do not like traditional, non interactive (we will call them 'frontal') lectures, where a teacher presents his topics to a public of 'listeners'. This system of 'transmission' of knowledge normally has a rather low level of effectiveness in every cultural domain, especially in primary and lower-secondary education.

    In pre-service and in-service mathematics teacher education, sometimes 'frontal' lectures may be very useful - if the audience is sufficiently motivated and 'frontal' lectures are part of a comprehensive program which includes other activities (small study groups, working groups, general discussions, brain storming sessions, practical experiences...); actually they can provide:

–  introduction and/or framing of problems, to be dealt with in working group sessions;

–  presentation of a theory in the field of didactics of mathematics (especially when theory meets the need of interpretation of important phenomena);

–  presentation of a new perspective about mathematical issues already known, or cultural connections between school mathematics and mathematician mathematics (producing a critical revision of school mathematics);

–  synthesis of general discussions or working group experiences.

In these cases, prospective (or in-service) teachers may appreciate (in consideration of their professional future) using 'frontal' lectures in a functional way (which obviously is not the only one). Actually the only experience of 'frontal' lectures most of teachers usually have is their own experience as students - where frontal lectures, followed by 'practical exercises' and 'evaluation tests', represented the standard organization of the 'transmission' of mathematical knowledge. If teachers experience an efficient teaching system where 'frontal' lectures are functionally included in a wider educational program, they may change their attitude towards 'frontal' lectures in their teaching activities.


4.3   By carrying out problem-solving activities prospective teachers directly experience what constructive learning means and what kind of difficulties it may involve


Here, we mean 'problem solving' in a broad sense, including: construction of conjectures and mathematical proofs, construction of definitions, formulation of an original survey about a topic, etc.

    Nowadays, problem-solving experiences are very common in teachers' pre-service and in-service mathematics teacher education. Many articles have been published in the last fifteen years on this subject, which show the potentialities of these activities from different points of view. Problem-solving activities for adults properly carried out (i.e., directed by a competent 'tutor') may help prospective or in-service teachers:

–  to experience and discuss difficulties similar to those met by students in the class;

–  to understand the importance of evaluating, in mathematical activities, the 'process' instead of the 'result';

–  to discover the cultural importance and working value of mathematical topics previously learned as specialized objects with no cultural perspective and retained as inert knowledge;

–  to understand what mathematical tools are not really mastered (quite often, prospective and in-service teachers believe they master some subjects only because they passed some examinations on them; this does not imply working mastery of those subjects as requested in problemsolving activities);

–  to find out where to apply epistemological, cognitive, 'didactical' tools - or where to experience the need of them.

The tools and results of research in didactics of mathematics that the methodology we are considering allows to introduce in mathematics teacher education, especially the 'qualitative information' and 'theoretical perspective' type results, can be focused under the guide of a competent tutor. Being personally involved in problem solving makes concepts like 'didactical contract', or 'epistemological obstacle', or 'didactical transposition' easier to understand. If the tutor carries out a cross analysis of problem solving strategies and solutions amongst the teachers, it is also possible to train them in the psychological and didactical analysis of the problem solving process.

    The general favour gained by the methodology we are dealing with, generally shared by mathematics educators, may hide some of its deficiencies and limits:

–  reasoning, motivations, relationships with the teacher may be different in adults and in young students;

–  the occasional application of some tools and results of research in didactics of mathematics during the analysis of problem solving strategies and difficulties met by adults does not allow a general view about theoretical framing and the relative importance of those tools and results in the field of research in didactics of mathematics;

–  the success of the education through this methodology heavily depends on the tutor's personal qualities: the ability to choose suitable problem situations (according to teachers' previous experience in the domain of mathematics), great facility in interacting and quickness in connecting different domains of knowledge in research in didactics of mathematics are needed.

Besides these deficiencies and limits, some institutional and cultural aspects may prevent the methodology we are dealing with from working properly: if the evaluation system of pre-service mathematics teacher education does not distinguish clearly the assessment of mathematical competence and the assessment of competence in the field of didactics of mathematics, it is very difficult to involve pre-service teachers in a constructive problem-solving activity and to get their cooperation in providing genuine materials concerning their problem-solving process. In-service mathematics teacher education experiences another kind of difficulties, when problem-solving activities are proposed to teachers: they may fear to compromise their image of 'competent' people-who never fail! And sometimes this fear does not concern only other people's estimation, but their own self-consideration, in the sense that they could not cope with failure. In this case, the tutor's role is very important and difficult: he/she should encourage teachers to run the risk of failing, carefully consider their difficulties, and suggest that their students might profit from a situation where a positive 'solution' is not ensured as the final, unavoidable consequence of a process driven by stereotyped models.


4.4   Discussing professional problems personally met in the class (during pre-service apprenticeship, or normal teaching experience)


This methodology is widely practiced in the world as a crucial aspect of selftraining of groups of school teachers; sometimes it takes place under the supervision of a coordinator (a school teacher with special competence in teacher education, a professional teachers' tutor or an university teacher). In any case, this methodology seems to be very useful to show the potentialities of planning of teaching, overcoming teaching difficulties and learning from experience as a collective task.

    Traditionally, the teacher (especially the mathematics teacher, whose subject frequently is not connected with the other subjects taught to the same class) acts under his personal responsibility as an 'isolated' entity in the school and may end up finding himself in a difficult personal situation, due to the special hardness of his subject and its value as a way to perform selection. In this perspective, discussing with colleagues about what happens in the class is a good opportunity to escape isolation and share responsibilities.

    Unfortunately, in many cases the lack of specific knowledge in mathematics education as a field of scientific investigation prevents from taking advantage of the potentialities of these peers' discussions. Especially when preservice teacher education in the domain of mathematics education is not proper, the discussion is nothing more than an 'episode'; at first this may be useful from a psychological point of view (in any case it gives the impression of sharing with others some heavy professional responsibilities!), but in the long run it becomes very difficult to share common problems and interpret them within a common frame, working out solutions together. Frustrations may come out from it, and the final idea of a 'waste of time' too. This is the reason why we think that teachers' working group concerning their professional problems must be supported by M.E. researchers in many ways (according to traditions, material and economic constraints, etc.): videotapes, booklets, education of tutors, direct involvement in the teachers' working groups.

    Taking into account our experiences, we think that the best solution is a direct (though not continuous) involvement of researchers in teachers' working concerning the problems they meet in their classrooms: this helps teachers make immediately contact with research tools and results, and offers researchers the opportunity to know teachers' problems and learn how to make their own language and perspectives understandable. This choice results in a mutual trust climate and a common language.

    In our experience, especially the results of the 'qualitative information' and 'theoretical perspective' types need researchers and teachers to be directly in contact in a working group to discuss teachers' professional problems. Other kinds of results, useful to support these teachers' group discussions, may be easily provided through booklets, articles, etc.


4.5   Discussing selected materials, videotapes, quantitative results, etc. resulting from research work or from other teachers' class experiences.


Videotape facilities and extensive research reports now available make this methodology of teacher education very popular all over the world. Distance teacher education is commonly based on that: the education of the tutor who coordinates peripheral group discussions seems to be relatively easy, due to the fact that discussions concern a pre-established set of written or video materials.

    This methodology should provide teachers with a quick, concrete, understandable access to tools and results of research in didactics of mathematics, especially of the 'innovative pattern' and 'qualitative information' types.

    However, the implementation of the Italian regional programs of compulsory in-service primary school teacher education at the beginning of the 90's shows some intrinsic limitations of this methodology. Our group was directly involved in the program for our region (both in making some videos, and tutor education). We tried to make videos showing class experiences, and provide some general guidelines to frame and interpret them from the point of view of research in didactics of mathematics (both as comments about what had happened in the class, and as lectures based on examples and students' class materials). Some teachers belonging to the Genoa Research Group in Mathematics Education directly took part in the peripheral tutoring program.

    The main education problem was to help in-service teachers get to know a complex field like research in didactics of mathematics through this methodology in order to get tools and results useful for their profession. In our expe riences, two facts emerged: on one side, teachers tended to get, from other people's experience, 'recipes' (regarding how to behave in the class on a specific subject, or innovative methodologies) rather than 'criteria of analysis' for what happened in the class. On the other side, teachers thought that some experiences did not concern them: 'I am able to prevent that problem from emerging', 'My students do not behave in such a way'. 'Those students are great, I never met such a class', 7 do not understand why to waste so much time in analyzing a thinking process that resulted in a failure'. Similar things were reported about similar education experiences carried out in other Italian regions by local research group in mathematics education. In the U.K. the Open University has done a lot of research on these beliefs.

    In conclusion, we think that the methodology we are dealing with may be very profitable (from the point of view of teacher education and inexpensive dissemination of tools and results of research in didactics of mathematics) when the methodologies considered at 4.3 and 4.4. are adopted as well, providing teachers with a preliminary, direct access to educational perspectives through the discussion of their professional problems and personal experiences.


4.6   As a conclusion concerning methodologies


In our opinion, each of methodologies described above gives some opportunities and has some intrinsic limitations in mathematics teacher education; a well balanced mixture of them is needed (according to different kinds of results of research in didactics of mathematics to be made available to teachers as well). And we think that such a mixture might also suggest that teachers should be flexible in organizing their own teaching activities!

    (as to our experiences, cfr. Belcastro, Guala & Parenti, 1986; Chiappini & Parenti, 1991; Chiappini, Laviosa & Parenti, 1990; for other countries, cfr. Borasi & al., 1994; Carpenter & Fennema, 1989; Cooney, 1994; Krainer, 1994; NCTM, 1994; Silver, 1994; for general discussion, see Cobb & al., 1990; Cooney, 1994; Houston, 1990; Hoyles, 1992; Lerman, 1990; Simon, 1994; Wittman, 1989, 1991).


CONCLUSION


In our opinion, results and tools of research in didactics of mathematics are needed both to study phenomena concerning mathematics teacher education (teachers' conceptions about their cultural and professional needs, failures and success of specific mathematics teacher education experiences, and so on: see 2. and 4.), and to increase teachers' professional performances (see 3.). Unfortunately, not always today research in didactics of mathematics is able to provide teacher education with the necessary tools (see 3.), and not always mathematics teacher education is sufficiently open towards research in didactics of mathematics - or conveniently organized from the methodological point of view (see 4.).

    As a consequence of this situation, we see that many opportunities are lost in both directions: the increase of the level of teacher vocational education, and the further development of research in didactics of mathematics. Local institutional and cultural aspects and traditions may create additional difficulties in some countries: as a result, tools and results of research in didactics of mathematics frequently remain outside the 'core' of mathematics teacher education, with a considerable waste of skills and energies.


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