1. Introduction
The interest addressed to teaching and learning processes, in particular to long-term processes, constitutes a feature which is peculiar to Italian research in Didactics. This is also due to the institutional specificity of the Italian school system which sees the Mathematics teacher with the same students for several years and to the peculiarity of our research groups which are connected to universities and include for common studies both university researchers and in-service school teachers.
In the following sections it is evident that some studies are mainly focused on teaching processes, others mainly on learning ones and the interlacing between these two types of processes is always present: it seems difficult to distinguish or isolate them, even at the level of a theoretical analysis. The question of the pertinence of such a distinction for research in Didactics, especially in reference to long term processes, is still open.
2. Social construction of mathematical knowledge
Over the last five years, numerous research projects have been developed and have focused specifically on the analysis and study of models of teaching and learning processes. Particular attention is paid in these research projects to the analysis of the social construction of knowledge, to the activities of negotiation in class and to the role of the teacher.
2.1 On verbalisation process
One of the main features of the Italian research in Didactics of Mathematics is a special emphasis on "verbalisation" which characterises both the teaching processes and the learning ones. Demonstrating, arguing, discussing and explaining, are crucial steps in doing mathematics: they are always present during didactical activities and are also requested from students at every scholastic level.
All that has very deep roots: it belongs to a particular cultural tradition and furthermore it is favoured by some features of our scholastic system, for instance, as just mentioned, the fact that the same teacher follows the same students for several years and the presence, in every class, of pupils of different levels of ability.
In 2.2 and 2.3 the function of the teacher in the process of a social construction of mathematical knowledge and the specific role of discussion will be detailes. Here we intend to stress the special impulse given to research on this theme by the group of Boero which had the merit to focus and to induce other research groups towards a particular attention on the development of verbal language as an instrument of thought. More precisely, verbal language is considered to be the heart, not only of the expression of mathematical concepts, procedures, reasoning and deductions, but also of the heuristic aspects of planning and solving processes for problems (Boero, 1989b, Lemut and Ferrero, 1988).
In reference to students aged 6-14 (our compulsory school period) research carried out by the same group highlighted that didactical interventions which have systematic recourse to tasks of verbalisation can be decisive in the construction of mathematical competence, also for pupils with learning difficulties (Boero, 1989a).
2.2 Verbal interaction in class
Arzarello, Bazzini and Chiappini (1994), characterise in systemic terms the resolution process of problematic situations by introducing the concept of the resolutive world. They define a model of the learning process in scholastic situations, based on their studies in relation to learning algebra.
A resolutive world is a world in which the subject produces an interpreting factor relative to the problematic situation proposes. This is characterised by at least the following factors: (A) the characteristics of the problematic situation; (B) the system of signs which is used as a mediator of thought and action; (C) the characteristics of the social interaction which, in such a world, develops in relation to the knowledge in play; (D) the fund of knowledge which the subject possesses; (E) the psychological behaviour of the subject who acts to construct an interpreting factor. A resolutive world is therefore defined by the mutual relationships which are formed between such components. Each student develops his resolutive action within his own resolutive world and at the same time contributes to the characterisation of the resolutive world of the people who he interacts with and viceversa. From this viewpoint, the authors confirm that the activity of negotiation in class can be particularly useful in the learning of algebra.
This theory is in accordance with the results of some Italian researches on social interaction, verbalisation, and formulation of hypotheses. Teaching and learning processes are analysed highlighting how such activities can supply relevant information regarding the identification of a didactic suited to the knowledge in play (Boero et al., in press, Bartolini Bussi, 1989, 1991)
The necessity to integrate epistemological analysis of knowledge with linguistic analysis is confirmed by studies in the area of the research project "Mathematic Discussion", conducted since 1986 by the group directed by Bartolini Bussi. Such a project, according to Bartolini Bussi's classification (1994), takes its' place among the innovation research projects, characterised by the theoretical choice to consider innovation in the class as an instrument and effect of a growing knowledge of the teaching and learning processes: the poles of action in the class and of the knowledge of teaching and learning processes are developed in a dialectic way. In the view of european tradition expressed by the studies of Vygotskij, in this research the student's learning process is not considered isolated but is studied and planned together with the teaching process used by the adult.
In Bartolini Bussi and Boni (1995), the theoretical reference picture is presented along with a detailed analysis of the activities of mathematic discussion on the concept of "point of view". In particular, the analysis instruments of verbal interaction in class used in the research are illustrates. Such instruments are based on a critical analysis of models introduced into the linguistic area by Leont'ev[1] and Lumbelli[2] and on a consequent theoretical re-elaboration directed towards the aims of the project.
The situations of verbal interaction in class, with particular reference to the activities of formulation and verification of hypotheses, are seperate subjects of study, dealt with by other research groups.
In Scali (1994), Boero et al. (to appear), Boero et al. (1994), formulation situations and verification of hypotheses relative to different disciplinary circles and scholastic levels are analyses. Characteristics connected to certain situations are pointed out, for example those regarding the learning of astronomy or the teaching of programming in the field of informatics, where the establishment of cognitive conflicts aids learning: in this case no doubt the role of the teacher is central, and the entire following paragraph is dedicated to that.
2.3 The role of the teacher
In these last years many Italian researchers have faced the problem of the role of the teacher in didactic situations. Here the dilemma is: when should the teacher support the "natural" development of the students' learning and when on the other hand should he/she force their learning?
Some papers underline the usefulness of a mixed strategy and many emphasise the crucial role that is embodied by the teacher in stimulating students' metacognitive abilities. In particular, Pellerey (1990) and Pellerey and Orio (1995) set out in detail the metacognitive activities which are indispensable to students in the phase of construction of their own knowledge and of the control of their own cognitive processes and the possible consequences regarding the results of learning are describes.
Moreover Martinelli et al. (1990) analyse the didactic importance of the teacher's choices with respect to the themes to be worked out in class (particularly in Italian compulsory school, because of its integrated curriculum of sciences and mathematics).They also describe the activities for the students and the way of managing pupils in class. Many concrete examples of such choices are pointed out in the paper dealing with pupils between the ages of 11 and 14.
Bartolini Bussi's research (quoted in the previous section) analyses the role of the teacher as the "orchestrator" of discussion in class: it also underlines the importance the teacher assumes in this sense, in an analysis of learning which examines processes in the long term.
A theoretical analysis of the role of the teacher in the class is dealt with in Boero et al. (1994): here the teacher is seen as a witness of culture elaborated by man in the course of history, who helps students in major tasks which are beyond their autonomous abilities; for example, the teacher can help them by offering different ways of looking at a phenomenon (particularly when they are in impasse situations), by supporting and enhancing their ways of representation of a situation (simplifying their grasping of the situation), by suggesting different ways of reasoning (allowing connections not spontaneously induced, which prove effective for the students to rule the complexity of the situation). The paper discusses what is called a 'personalised mediation method' in technological environments, that is a method which is aimed at making the pupil take into account objective constraints of the situation: for example at elementary school, helping him/her by lending the words necessary to enable him/her to accomplish his/her spoken intention, making the pupil dictate his agreement in phrases to his/her 'scribe-teacher', sustaining the production processes of oral texts and self-dictation of the gradually constructed phrases. The research emphasises the effective growth in individual capabilities whenever the teacher's intervention affects the procedures used by the child to answer the question given.
Particularly, the evolution of pupils' reasoning while producing hypotheses concerning technlogical processes is screened, considering the material gathered in the classroom (i.e. texts produced by pupils, records of interaction, etc.). The results found consist of the internalisation of the logical components of the reasonings made during shared social activities, as well as of the hypotheses on technological processes. With reference to the fields of experience of technological interest, the so-called game of hypotheses (Boero et al., in press) seems crucial in the transition from the field of experience as an early learning environment fostering the teacher's mediation as well as the child's first meaningful activities, to the field of experience as a cultural environment where the pupil exerts his/her intelectual mastery and is allowed to perform in a more articulate and complex manner. The evolution from opinions to interpretative and planning hypothesis appears to be favoured by the fact of working on common experiences shared by all children and grounded on material constraints. The authors claim that in order to force the gradually more independent, active and deep adhesion to the game of hypotheses, it is not sufficient to activate socio-cognitive conflicts (Bartolini Bussi, 1991) in the class on hypotheses as such. The technological objects appear to build necessary, external objective references, especially in the validation phase.
Analogous claims are supported in Arzarello, Bazzini, Chiappini (1995) where the authors say that such major abilities as anticipating and planning in algebric problem solving (see also the comments on Boero, in press) can be developed only within a suitable 'didactical space-time of communication and production' (SP in short, see 3. for more details), where students can find both social interaction, that is interpersonal exchanges between the pupil and the environment (teachers, mates, ...) and suitable mediators (namely cultural artefacts, like books, computers, ...) which are aimed at producing meanings. To be useful for learning, the activities done by pupils must allow them to validate and justify the sense of the expressions within their social space, that is within the rules and the culture that they effectively own. All this constitutes the socially shared, even if not explicit, background which belongs to the social space of the pupils and constitutes the necessary basis for what pupils do and for their interpretations of what they have done. SP must be able to scaffold and support the student's activity; the learning process can be considered as an imitation, on the part of the novice, of the expert's performances: from his/her side, the expert can support the novice by explaining his/her own strategies, streams of thought, skipped difficulties, while solving a problem.
The social aspects of interactions are essential to scaffold pupils' activity as a 'cognitive apprenticeship' (Arzarello et al., 1993) and not only as a practical one: in such teaching-learning environments the novice's social space grows also because he/she learns by doing, by seeing it done and by systematically discussing what he/she is doing both with experts and with other novices.
2.4 Theory of the situations
Some research work has used and/or critically analysed the model of the theory of situations, introduced by Guy Brousseau[3] and developed by the French school of mathematics didactics.
In particular, Arzarello (1994) considers the theory of situations as the most appropriate one to the modelling of learning processes regarding the construction of concepts. Such confirmation is the result of studies on the learning of algebra where the duality of methods is highlighted: construction of concepts and learning of mechanisms. According to the author, both plans run serious risks, each dual to the other: the first one places too much hope in the automatic conforming of symbolic language to concepts considered the keystone of learning; the second one risks reducing learning to a pure question of signs, hoping in a so-called natural acquisition of the concepts. The automatisms are based on aspects of socialisation and institutionalisation in the first case, whereas in the second case one hopes for a natural conforming of the ideas to signs which it would have in its' own private language. In both cases, the construction of the sense of the mathematic symbol i.e. of the link between signs and concepts, has been disregarded, risking an unbalanced approach to mathematics.
Polo (1993), proposes a critical analysis of a communication situation, carried out in 3rd and 4th French and Italian primary school classes. The activity concerns abilities relative to spatial organisation and to the identification and use of reference systems which are external to the subject and pertinent to the description of physical objects. The analysis uses concepts bearing the theory of situations with a double aim. On the one hand, to tackle the study of the possibility of reproducing an experimental device tested in the French school system, in a normal scholastic situation in an Italian class. On the other hand, to highlight the conditions (or variables) in the analysis prior to the situation, which allow the situation to maintain the a-didactic character, compared to the reference system considered as knowledge in construction. In particular, the context of team play introduced in the experimental device, but not indispensable from the point of view of the functioning of the knowledge in play, was not maintained in the activity proposed in the Italian class, who were well used to group work and problematic situations. Furthermore, in relation to the results of Berthelot-Salin[4], it was highlighted as an activity connected to the space of the physical world, not in itself sufficient to lead the subject to reason according to a problem of a geometric type and so calling on geometric knowledge.
Bianchi, Pertichino and Piochi(to appear) discuss the possibility and the pertinence of using the concept of a-didactic situations to recuperate basic concepts in the first year of high school, paying particular attention to special cases of difficulty in the learning of mathematics. This work describes an experiment carried out in the 1st class of a Professional Institute, the field of the experiment is that of economics and the mathematical concepts dealt with range from reading, writing, comparison and operations with large numbers, to ratios, proportions, fractions, reductions and percentages. The results obtained from the point of view of the popularity of the activity as far as the students were concerned, are, according to the authors, an encouraging sign to the research of analogous situations for other mathematical subjects.
3. Concepts, signs and representations in Mathematical Knowledge construction
The relationship between the concepts and the signs which are used to represent them and the consequent problems from a didactic point of view have been dealt with by some researchers. There are two major types of research projects, namely those where algebraic and/or pre-algebraic concepts are involved and those concerning geometrical notions and representations.
As far as the former is concerned, the interplay between representations (i.e. drawings) and concepts is studied in various ways. This wide area of research, which has clear and distinctive features, is thoroughly analysed in the section edited by Bartolini Bussi and Mariotti, which we make reference to.
We deal here with research where algebraic language is involved: the aim is to scientifically define how the mastering of concepts and signs can be achieved in an effective way, for example through situated problem solving or in other ways. The thinking processes of pupils engaged in problem-solving, where syntactical manipulations are necessary to cope with the situation, are analysed in order to get information on what happens in the class when such tasks are faces.
With more details, Boero (1995) analyses some crucial strategies in algebraic problem-solving, where the main points consist of anticipating forms of simplification of the task and its' resolution and in consequently transforming the problem in order to manage it better.
The process of transformation may happen without, before and/or after algebraic formalisation. When the transformation happens without or before algebraic formalisation, the process is ruled by arithmetic or geometric, or physical manipulation of variables (pre-algebraic strategies). When the transformation happens after algebraic formalisation, it is often based upon the transformation function of the algebraic code, which enormously extends the range of possibilities of the transformation. The process of transformation calls for specific prerequisites and skills, for example the mastery of standard patterns of transformation.
A common ingredient to all processes of transformation is anticipation. In order to direct the transformation in an efficient way, the subject needs to foresee some aspects of the final shape of the object to be transformed related to the goal to be reached, and some possibilities of transformation. This anticipation allows planning and continuous feed-back. In the case of transformation performed after formalisation, anticipation is based on some peculiar properties of the external algebraic representation.
A consequent didactical problem concerns which educational strategies are suitable for enhancing the development of the anticipation processes. (On this point see also below the comments on Arzarello, Bazzini, Chiappini, 1995).
An analogous approach to the mental processes of pupils engaged in transformations of representations (in algebra as well as in other fields) is taken by Gallo and her research group (Gallo et al., 1991a, 1991b, Gallo, 1993, 1994). The main question here is the control during the solution process of a problem (for example the factorisation of an algebraic formula, but also a geometric problem); such a process can be thought of as a chain of cycles: the pupil passes from one mental model to another, hence producing a sequence of models; to work better with the problem, through control, the pupil transforms the problem and builds up suitable thinking objects, namely transient solutions. This process of going back and forth from the mental models to the transient ones (descending control) and vice-versa (ascending control) ends when the pupil produces the final model, which is the solution for him/her (and which may or may not be the real solution to the problem). The main obstacle to the development of this process is the rigidity of the models which students use and the consequent major didactical problem in the building up of dynamical situations where the control is sustained by the situation itself. In the quoted papers Gallo and her collaborators have developed an analysis and classification of various exercises and problems in algebra as well as geometry, when the solution processes are analysed from the point of view of ascending and descending control.
The processes of pupils engaged in algebric problem solving are also analysed by Arzarello, Bazzini, Chiappini (1994, 1995): they describe a theoretical model suitable for analysing the cognitive processes of students involved in algebraic problem-solving. It is based on the distinction between sense and denotation of an algebraic expression: the two notions are taken from Frege's old semantics, which has guided all the modern research on the meaning of signs.
The dialectic between the two develops within suitable conceptual frames, switched on by pupils while solving algebraic problems. Their strategies can be described as a game of interpretation from one frame to the other, stirred by the unbalanced roles that the sense(s) and the denotation(s) of formulas assume with respect to the different frames.
Two notions are essential to describe the game. The first is the naming process, that is when pupils puts ideas into formulas, at the very beginning of the game of interpretation. The second consists of the switching from one interpretation to the other in possibly different conceptual frames: its' success is marked by processes of condensation of concepts in synthetic formulas and acts of thought, whilst poor performances are accompanied by the symmetric process of evaporation, which happens when pupils lose the complex relationships between the senses and the denotations of their formulas and remain with their syntactic aspects as the only owned meaning.
The consequent didactical problem is the building up of situations which sustain the pupils'game of interpretation; this problem has been dealt with in Arzarello, Bazzini, Chiappini (1995) where they introduce the notion of 'didactic space-time of production and communication' (SP in short, as just mentioned in 2.3). It is a space designed by the teacher in order to allow the learner to plan activities, based on a task.
The main goal of the student's plan is the accomplishment of the task; the teacher's goal is to put the student in touch with the knowledge to be learnt, through the designed activity. In this frame, the learning of knowledge can be developed only through an activity: the meaning attached to it always depends upon the functionality of the semiotic system used within the activity itself. Hence the concrete process of learning consists of using the semiotic system's functionality within the SP, in order to incorporate the senses which are switched on and shared through the activity, achieving the meaning of the situation in the end.
The word space means the frame within which the subject's activity can develop as a production-communication activity. An SP is defined by the possibilities of action that are available within it and by the specific features and modalities after which the subject's actions can be realised as production and communication activity. The SP supplies the task, suggests the context and provides the tools, both for the action and for a socially shareable meaning of the action's product. An SP is able to create a social space and is useful for the student's learning if it can scaffold the activity of the pupil with respect to the knowledge to be learnt in a way that is accessible to the evoked social space. An SP allows us to reproduce for didactic purposes (i.e. not in a spontaneous process) the crucial match between a support system in the social environment and an acquisition process in the learner.
An SP is meaningful for the learning of knowledge, provided that the possibilities and the modalities of action, which it allows the pupils, realise the following goals: (1) motivating pupils for their planning of the activity; (2) supporting them in the choice of the specific goals and in the consequent processes of anticipation and planning; (3) emphasising the functionality of the system of signs into which the knowledge to be learnt must be incorporated, according to the design of the teacher.
The first two goals are a necessary, but not sufficient, condition for achieving the planned learning: they can be pursued by means of adjustment processes coached by the subject with respect to the possibilities and modalities of action within the SP. The third goal requires pupils to acknowledge such a functionality, and the fact that they have a concrete accessibility to it within the SP: this means that pupils are proximal (in the sense of Vygotsky) to convert the represented knowledge from one known system of signs to another one and that the activity designed in the SP can support and help them in achieving such a goal.
The theme of the conscious passage from one system of representation to another was dealt with in specific different contexts by the research group co-ordinated by Pesci. Their studies, which concur with some theoretical lines of thinking on this theme (for example the theory of the "concrete carriers" by Dörfler[5]), were developed starting from the following hypotheses:
More precisely, the graphic representations examined are: the tree graph, in reference to probability in the case of random compound events (Pesci, 1994); the arrows diagram in "inverse" problems, both as a means to facilitate the phases of comprehension and solution of an inverse problem and as images which in any case aid the comprehension of the concept of inverse function (Pesci, 1991, 1995); the signs produced by the student to represent multiplicative situations before and after the study in class of proportional reasoning (Pesci, in press).
The researchers also point out the possibility of a specific didactic aimed at developing students' "metareflections" on graphics symbols produced by themselves or already used on a regular basis (Bertolini et al., 1993, Giuliani et al., 1993, Baldrighi et al., 1994, Cavallari et al., 1994).
4. Analogies and obstacles as learning processes models
In the context of a concept of learning as an active and constructive process, the idea of analogy, which consists, in an extreme synthesis, of the matching of two areas of knowledge, covers a very particular area. Some studies have recognised, for example, the function carried out by analogy in the recuperation of information stored in the memory, in the coding of new information and in the restructuring of knowledge already acquires. It is noted, furthermore, that reasoning through analogies aids the formulation of hypotheses, even though it may be the cause of misconceptions and misunderstandings. Even research into the didactics of mathematics has highlighted interest in this theme, we are referring to, for example, the contribution of Polya[6] or that of Fischbein[7] .
Setting out from this starting point, the research group co-ordinated by Bazzini thoroughly investigated this theme and in reference to the primary school, elaborated and experimented some didactic units with the aim of stimulating the recourse to analogical thought in its' most productive form.
Having identified the analogy in a theoretical model which refers to the fregean distinction between sense and denotation of a certain expression, Bazzini (1995) also presents the theoretical analysis of some didactic units (for children aged 6 to 9) regarding the construction of numerical knowledge, in the hypothesis that analogical reasoning on the one hand calls for but on the other stimulates mental flexibility in favour of learning.
A model of a learning process which is connected to the analogy but which focuses on its distorted application is that which revolves around the concept of epistemologic obstacles described by Marino and Spagnolo(to appear). Their main concern is twofold: examining the notion of obstacle, as it has been defined in the literature, particularly focusing on its' features from the point of view of a communication theory, and analysing mathematics as a language, namely with respect to its' syntax, semantics and pragmatics, in order to show its' relationships with epistemological obstacles and also a possible way of facing them. The postulate of Eudoxus-Archimedes is taken into consideration and an experimental study, designed to test if the postulate is an epistemological obstacle, is described in the quoted work together with the ways students tackle such a postulate and the continuity of the line.
5. Technological media for a meaningful learning
In addition to what has just been said in previous section about the use of technological media to promote students' cognitive processes, this paragraph deals with specific studies carried out on this theme during the last few years.
More precisely, Di Carlo and Trentin have studied possible uses of formal languages in mathematics didactics with the aim of promoting learners' activity, especially in the process of own knowledge restructuring. They describe (1990) a cyclic learning-process model upon which didactical experiences for 14-15 year old pupils have been plannes. It is during the phase of formalisation and organisation of new contents that students may find difficulties and therefore the majority of them passively accept guidance from teachers or text-books. In this phase it is interesting to intervene with proposals which promote students' internal assessment and awarness: giving them the task of producing original diagnostic tests on specified contents, acting as authors, seems to be a successful way for that. To develop this material on computer the DELFI system, which is based upon specific methodology for diagnostic tests, was chosen and the mathematical subject taken into consideration was the total or partial order relation. The student authors had to develop a net of connections between concepts and information involved, tackling the work in groups and critically discussing every step of the proposed activity. The two final phases of the described experiment were the proposal of these tests to other classes and the debate between users and authors about the material produces.
The same authors describe another experience (1989) more orientated towards problem-solution but still aiming to promote the student's active participation. In this paper the reference is to Petri net, a formal language to describe and analyse knowledge domains. A Petri net is an orientated graph where two types of nodes are essential: those which refer to required skills and knowledge and those which refer to activities necessary to solve a particular problem. Several examples are detailed and the different steps of the proposed activity are describes. Also in this case positive results were obtained in reference to class participation, students'awareness of their mathematical specific knowledge and the competence in using their skills.
The idea of the importance of cooperative work, especially in the phase of self-evaluation, is central also for Di Carlo et al. (1993) which present as a case study the inequalities of first and second grade: many frequent errors are considered and possible ways to discard them are describes.
Computer based learning environments are considered by Farinetti and Scarafiotti (1995) software tools which can create an interactive dialogue between student and teacher. In particular they describe a specific model on which the construction of a hypertext, hypermath, is based, to learn calculus through the solution of problems: it is addressed to students in the first year of a masters degree in engineering. The basic idea is that the non-linearity of hypertexts can help learners in the process of assembling and disassembling the logical and temporal sequence of their own thought. The proposed hypertext provides two different routes of navigation, one according to a guided model of exercises, the other according to a non-guided one: in each of them, the student has the possibility to select different routes to the solution of the problem, also taking into account the opportunity to use the botton 'analogy' which may concern the formulation, the subject or the procedure of the problem.
Bibliography
Arzarello, F.: 1992, Algebraic problem solving, Mendes da Ponte, J.P. et al. (eds.), Mathematical Problem Solving and New Information Technologies, NATO ASI Series F, Springer, 89, 155-166.
Arzarello, F.: 1994, L'apprendistato al senso dei simboli in algebra,s L'insegnamento della matematica e delle Scienze integrate, 17A-17B, n.5, 536-554.
Arzarello, F., Chiappini, G. P., Lemut E., Malara, N. A. and Pellerey M.: 1993, Learning programming as a cognitive apprenticeship through conflicts, Lemut, E., du Boulay, B. and Dettori, G. (Eds.), Cognitive models and intelligent environments for learning programming, Nato ASI Series F, 111, 284-298.
Arzarello, F., Bazzini, L. and Chiappini, G. P.: 1994, L'algebra come strumento di pensiero, Analisi teorica e considerazioni didattiche, CNR-TID project, IDM series, 6.
Arzarello, F., Bazzini, L. and Chiappini, G. P.: 1995, The construction of algebraic Knowledge: towards a socio-cultural theory and practice, Proc. PME 19, Recife, 1, 119-134.
Baldrighi, A., Giuliani, E., Joo, C., Pesci, A. and Romanoni, C.: 1994, Ratio concept and graphical mediators: an exploratory study with 13-14 year old pupils, Proc. CIEAEM 45, Cagliari, 1993, 92-97.
Bartolini Bussi, M.: 1991, Social Interaction and Mathematical Knowledge, Proc.PME 15, 1, Assisi, 1-16.
Bartolini Bussi, M.: 1992, Mathematics Knowledge as a Collective Enterprise, Seeger, F. and Steinbrig, H. (eds.), The Dialogue between Theory and Practice in Mathematics Education: Overcoming the Broadcast Metaphor, Materialien und Studien Band 38, IDM Bielefeld, 121-151.
Bartolini Bussi, M.: 1989, Evaluation of Teaching Sequences that include individual and collective Activities: Two Case Studies, Bazzini, L. and Steiner, H. G. (eds.), Proc.of the First Bilateral Symposium for the Furtherance of Scientific Exchange and Cooperation in Didactics of Mathematics, Quaderno CNR, 285-308.
Bartolini Bussi, M.: 1994, Theoretical and Empirical Approaches to Classroom Interaction, Biehler, Scolz, Strasser and Winckelmann (Eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer Academic Publishers, Dordrecht, 121-132.
Bartolini Bussi, M.: 1995, Tony and Dennis: Analysis of Classroom Interaction Discourse, Proc.PME 19, Recife, 1, 95-101.
Bartolini Bussi M.:(to appear), Coordination of Spatial perspectives: An Illustrative Example of Internalization of Strategies in Real Life Drawing, The Journal of Mathematical Beahavior .
Bartolini Bussi, M.: (to appear), Social Interaction and Mathematical Knowledge: Foreword, The Journal of Mathematical Behavior.
Bartolini Bussi, M. and Boni, M.: 1995, Analisi dell'interazione verbale nella discussione matematica: un approccio Vygotskiano, L'iInsegnamento della Matematica e delle Scienze Integrate, 18, 221-256.
Bazzini, L.: 1995, Il pensiero analogico nell'apprendimento della matematica: considerazioni teoriche e didattiche, L'Insegnamento della Matematica e delle Scienze Integrate, 18A, n.2, 108-129.
Bertolini, C., Maggi, M., Pesci, A. and Trevisani, M.: 1993, Un test esplorativo sul segno di uguaglianza in terza elementare, Atti Matematica e difficoltà n. 3, Pitagora Es., 73-80.
Bianchi,M. P., Pertichino, M. and Piochi, B.: to appear, Recupero di concetti di base nel primo anno di scuola superiore attraverso la proposta di situazioni a-didattiche, L'educazione Matematica.
Boero, P.: 1989a, Alunni con difficoltà di apprendimento: che fare?, Notiziario U.M.I., Suppl. al n. 3, 63-80.
Boero, P.: 1989b, Mathematical literacy for all: experiences and problems, Proc.PME 13, Paris, 1, 62-76.
Boero, P.:to appear, Transformation and anticipation as key processes in algebraic problem solving, Sutherland R. (ed.), Algebraic Processes and Structures, Kluwer.
Boero, P. and Bondesan, M. G.: 1993, Assessing mathematical potentialities in the "Zone of Proximal development", Proc.CIEAEM 45, Cagliari, 110-115.
Boero, P., Ferrari, P. L., Shapiro, L. and Ferrero, E.: to appear, Some aspects regarding the social construction of hypotheses in primary school mathematics classes, The Journal of mathematical Behaviour.
Boero, P., Carlucci, A., Chiappini, G., Ferrero, E. and Lemut, E.: 1994, Pupils' cognitive development trough technological experiences mediated by teacher, J. Wright and D. Benzie (eds), Exploring a new partnership: children, teachers and technology, A-58, Elsevier Science B.V.
Cavallari, A., De Angelis, A., Pesci A. and Toma, D.: 1994, Il segno di uguaglianza in ambito aritmetico -algebrico: attività per esplorare stereotipie e fraintendimenti, in Basso, M. et al. (eds.), Numeri e Proprietà, Università di Parma, 43-48.
Di Carlo, A., Accomazzo, P. and Scarafiotti, A.: 1992, Overcoming obstacles in concepts development: inequalities as a case study with the aid of multimedia, A.I. Weinzweig, Astrida Cirulis (eds.), Proc.CIEAEM 44, Chicago, 180-192.
Di Carlo, A., Garassino, M. C. and Rovero, G.: 1993, La verification formative pour irienter tout le procédé de construction de la connaissance. Les inéquations comme exemple d'étude, Proc. CIEAEM 45, Cagliari, 124-135.
Di Carlo, A. and Trentin, G.: 1989, Uso delle reti di Petri nella didattica della matematica, Notiziario U.M.I., Suppl. al n. 7, 122-139.
Di Carlo, A. and Trentin, G.: 1990, Diagnostic testing in formative assessment: the students as test developers, Bell C. and Harris D. (eds.), Assessment and Evaluation, Nichols Publishing, New York, 175-187.
Farinetti, L. and Scarafiotti, A. R.: 1995, Conoscenza e comunicazione nella costruzione di ipertesti per la didattica, Sistemi Intelligenti, Il Mulino, Bologna, n.2.
Farinetti, L. and Scarafiotti, A. R.: 1995, A strategy for flexible learning in mathematics, Proc.WCCE'95, Sixth IFIP World Conference Computers in Education, Birmingham, 23-28 July 1995.
Farinetti, L. and Scarafiotti, A. R.:to appear, Learning environments: cognitive bases and design problems, CC-AI, The Journal for the Integrated Study of Artificial Intelligence Cognitive science and Applied Epistemology.
Farinetti, L. and Scarafiotti, A. R.: 1995, Individual and group education: a methodological and experimental research, Proc.ECER 95, European Conference on Education Research, Bath, 14-17 September 1995.
Gallo, E.: 1993, Le contrôle dans la résolution de problèmes: une situation de classe, Proc.CIEAEM 44, Chicago.
Gallo, E.: 1994, Elaboration of models for problem solution in interaction with 14/15 year old pupils, Proc.Second Italian - German Bilateral Symposium on Didactics of Mathematics, Materialien und Studien Band 39, IDM Bielefeld, 289-301.
Gallo, E., Amoretti, C. and Testa, C.: 1991a, Utilisation de modèles géometriques en situation de résolution de problèmes: contrôle descendant et ascendant, Proc.CIEAEM 41, Bruxelles, 449-455.
Gallo, E., Battù, M. and Testa, C.: 1991b, The control in problem resolution, Proc.PME XV, Assisi.
Gallo, E. and Testa, C.: 1991, Modèles, stratégies, types de contrôle dans la résolution d'un problème graphique de géometrie, Tetradia Didaktikes Ton Mathematikon, 8, 55-78.
Giuliani, E., Pesci, A. and Romanoni, C.: 1993, Un'esperienza di avvio alla simbolizzazione in prima media, La Matematica e la sua Didattica, 1, 21-38.
Lemut, E. and Ferrero, E.: 1988, Of the linguistic prerequisites of computing literacy, Lovis, F. and Tagg, E. D. (eds.), Computers in Education, Elsevier Science Publishers B. V.
Marino, T. and Spagnolo, F.:to appear, Gli ostacoli epistemologici: come si individuano e come si utilizzano nella ricerca in didattica della matematica, L'iInsegnamento della Matematica e delle Scienze Integrate.
Martinelli, A. M., Boero, P. and Garuti, R.: 1988, The figure of the teacher as the promoter and organizer of his student' metacognition, Proc.CIEAEM 42, 244-252.
Pellerey, M.: 1990, Controllo e autocontrollo nell'apprendimento scolastico: Il gioco tra regolazione interna ed esterna, Orientamenti Pedagogici, 3, 473-491.
Pellerey, M. and Orio, F.: 1995, La diagnosi delle strategie cognitive, affettive e motivazionali nell'apprendimento scolastico. Costruzione, validazione e standardizzazione di un questionario di autovalutazione, Orientamenti Pedagogici, 4, 683-726.
Pesci, A.: 1991, Problemi inversi e schemi a frecce in prima media, L'Insegnamento della Matematica e delle Scienze Integrate, vol. 14, 2, 153-181.
Pesci, A.: 1994, Tree graphs: visual aids in casual compound events, Proc.PME XVIII, Lisbon, IV, 25-32.
Pesci, A.: 1995, Visualization in Mathematics and graphical mediators: an experience with 11-12 year old pupils, in R. J. Sutherland, J. Mason (eds.), Exploiting Mental Imagery with Computers in Mathematics Education, Nato ASI Series F/138, 34-51.
Pesci, A.:to appear, Using Graphical mediators for mathematical concepts constructions: the case of proportionality, in Rhodes, S. (ed.), Mathematics and Imagery.
Polo, M.: 1995, Interrélations dans un travail de recherce en didactique: une question pour le chercheur, Les Debats de Didactique des Mathématiques, Travaux et thèses de didactique, La penséè Sauvage, Grenoble, 21-34,
Scali, E.: to appear, Le role du dessin dans la modelisation géométrique élémentaire des phénomènes astronomiques, Proc.CIEAEM 46, Toulouse, 1994.
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