1. Introduction
The Italian research on the subject "Mathematics and Difficulties" has increased considerably, especially over the last few years, characterized by the awakening of researchers to this type of problems and by the increase in the number of conventions on this topic. For sure, one of the causes has been the enforcement of outline law ndeg.104 dated 5/2/92 on "the assistance, social integration and rights of handicapped people". In particular, article 12 refers to the right to education of the handicapped in nursery school, in the common classes of any type and level of school institutions and in university institutions, too". Paragraph 4 reads: "Learning difficulties or other difficulties resulting from handicap-related disabilities cannot be a hindrance to the assertion of one's right to education". In actual fact, this law has placed the learning subject in the centre of the educational process and, from that moment, the task of schools (and more in particular of educationists) has been that of fostering the cognitive, social and personality development through disciplinary contents. Moreover, this law provides for the assignment of specialized teaching staff with the object of ensuring recovery, socialization and integration. In particular, article 14 states that the Ministry has to provide for the training and updating of the teaching staff, as regards the acquisition of knowledge on school integration.
The outline law dated `92 is conclusive, as regards the regulations that started the process of school integration of the handicapped with law 118 dated 1971. This was not a law to reform school regulations, but simply a legislative act in favour of civil invalids and cripples. However, it has become part of the history of Italian school, as the first official legal pronouncement in favour of the insertion of the handicapped in normal schools. In fact, article 28 of this law ensures some provisions, such as free transportation from house to school and access to school seats by knocking down architectural features. Moreover, it asserts that "compulsory education must take place in the normal classes of state school". Afterwards, the integration process was legally improved with law ndeg.517 dated 1977 which undoubtedly is the most important legislative act on this subject, and one of the most advanced on an international scale. This law brought about substantial modifications to the regulations of compulsory school, as regards the evaluation of pupils, the abolition of resits and, above all, of differential classes (the provisions regarding integration are included in articles 2 and 7). Therefore, the implementation of this law means the insertion of handicapped pupils in normal classes, both in primary and secondary schools. In the course of the years, the regulations have been enriched and improved with the issuing of ministerial memoranda and sentences of the Constitutional Court. In particular, we wish to point out that in 1988, in consequence of sentence ndeg. 215 dated 1987, the Ministry of Education (with memorandum ndeg.262) gave directions how to put into practice the right to study of handicapped pupils, no matter what type of handicap, in any type and level of school, thus including secondary schools (that is the school attended from 14 years onward). Everybody knows that, with a document approved in 1990, the ministers of the European Community urged all of the member states to intensify their efforts towards the complete integration of the handicapped in school reality. We think we have explained how, in a certain sense, the Italian legislation has anticipated this exhortation. However, it should be noted that outline laws have rarely been followed by the administrative and economic provisions that would have allowed their proper enforcement. Attempts have been made to set up structures for the enforcement of these laws, but on a local scale, only. In fact, local situations show a greater sensitivity towards this type of problems. Results have been good, but ephemeral and, therefore, not really significant.
2. The Italian research
The didactic research groups engaged on this subject are those in Bari, Bologna, Cagliari, Florence, Genoa, Milan, Parma, Pavia, Pisa, Siena and Turin.
The decision to concern themselves with "Mathematics and difficulties" has been suggested by a particular sensitiveness to this subject, in some cases. In other cases, the stimulus has sprung from the involvement of researchers, in their capacity as teachers, in the specialization schools for the training of remedial teachers. These schools (which are usually two-year and apply selective entry) were instituted to give a suitable training, both from the point of view of specific disciplines in the classical and scientific fields, and from the broader standpoint of the knowledge of the problems relating to handicap. Sometimes, the need to tackle specific problems has directed research towards the drawing up and the study of didactic routes aimed at the achievement of general objectives, starting from mathematical activities. Other times, the acquisition of clear mathematical concepts, which is important in order to allow handicapped subjects to conquer their independence, has been chosen as the working subject. This has led to the working out of didactic units[1] that tend to stimulate and enhance the rational capabilities of subjects, succeeding in achieving the specific targets fixed, at the same time.
It is worth underlining that the research in this field is relatively "young" and, even if it has not a tradition behind it, yet, it is already bearing good fruits: for example, the activities proposed end by having important spin-offs on the usual didactic practice, even if they often diverge from it. In other words, it often happens that routes studied for pupils with learning difficulties become an integral part of general didactic programming.
A large part of the results of researches and experiments has been presented at the national Conventions on "Mathematics and difficulties", which are held every year in Castel San Pietro Terme (Bo). These meetings have been instituted to encourage the exchange of opinions, debates among researchers and between researchers and teachers. They also aim at spreading the researches on this subject. Everybody knows that one of the problems is just the spin-off of research results on school practice.
These are the topics that have been dealt with in these Conventions, up to now:
The Proceedings of these conventions have been published by Pitagora Editrice S.r.l., via del Legatore 3, I-40138 Bologna.
Other contributions can be found in the main Italian magazines dealing with didactics (see the bibliography).
As we have already said, the researches relating to the subject "Mathematics and difficulties" are rather recent and, therefore, classifying them is quite difficult. However, three research threads can be singled out, which regard respectively a) handicaps, b) social disadvantages, c) learning difficulties.
A basic goal of the mathematical activities proposed is giving subjects the possibility of having and using means for interpreting reality and intervening on it, critically. Therefore, mathematics has become an instrument for school integration and for the development of the handicapped's potential, as far as communication, relations and socialization are concerned.
Generally speaking, the researches on handicap refer to experiences with a single individual, with a specific didactic route aimed at the achievement of a fixed target and/or at the acquisition of a well-defined concept. The subjects tackled are connected with the role of mathematics as an instrument for individual growth towards the conquest of independence, that is the ability to get in touch with people or things and to interact with them. However, once this kind of choice has been made, traditional subjects, such as arithmetic, purchase and sale problems, watch reading, ..., cannot be left out, but these are not the only ones to be developed. We are aware of the fact that objects and subjects that are apparently far from school practice can be used in an advantageous way, to carry out important mathematical activities, provided they belong to the background of pupils and, above all, motivate them.
Of course, at the base of the research on the learning of handicapped pupils there is always the acquisition of material relating to the nature, the characteristics of handicap and the anamnesis of subjects. Moreover, the knowledge of the present theories of educational psychology cannot be excluded. After that, attempts are made to educate subjects to be the protagonists of their own cognitive activities, giving them external stimuli that involve people to the point that they are forced to resort to all of their resources. Of course, this type of choice implies the working out of routes susceptible of continuous modifications, brought about according to the outputs of the subject in reply to the inputs given by the researcher. The dynamism of routes facilitates the overcoming of stalemates and, in some cases, allows the achievement of positive results. In fact, it succeeds in awakening the devotion to work in subjects, and sometimes it leads to a significant learning. Moreover, it has been noticed that often these objectives can be achieved more easily if the route involves the whole class group and, therefore, it is carried out in classrooms, with the terms and conditions required by the subject's potential, of course. It has been noticed that the insertion in common classes stimulates behaviours of mutual collaboration, especially in compulsory school: in fact, a handicapped pupil is motivated by the desire to adapt his performances to those of the other pupils. The normally intelligent pupil feels fully conscious of his role of tutor and acts as a mediator between the teacher and the subject with difficulties.
One of the most recent trends of research concerns the problems relating to the design, making and use of software helps for disabled and, in particular, of those intended for pupils with a cognitive deficit (see the report "Thecnology and Mathematics", paragraph 2.6, in this volume).
Broadly speaking, the pupils who find themselves in these types of disadvantaged situations enter school without a clinical diagnosis and this is why it takes a longer time to recognize their difficulties. Moreover, no type of support is planned for this type of deficit and, therefore, these subjects face up to difficult and eventful school routes, and they often end by falling into the marsh of failure, which frequently drives them to give up their studies.
The children who belong to lower social classes find themselves in a condition of cultural lack, which chiefly shows with a linguistic deficit: they have a poor and inadequate linguistic heritage, with consequent difficulties in symbolizing, abstracting and conceptualizing. It is easy to guess that these difficulties play an important role in the learning of mathematics, which often is the discipline where the worst efficiency is joined to the most difficult recovery.
Some researchers have concerned themselves just with this type of difficulties, trying to work out didactic routes aimed, in the first instance, at motivating the interest for the subject, by generating adequate problematic situations. The starting point is undoubtedly the enhancement of cultural diversity. Consequently, the didactic interventions are not aimed at filling in a cultural deficit, but at strengthening personal experience, no matter what social class individuals belong to. The didactic situations proposed have privileged the channel of socialization and collaboration inside the "class" group, allowing, through the interaction with pupils, the achievement of some fundamental goals in mathematical learning, such as the concept of number, the positional value of digits and the mastery of the writing procedures necessary to make calculations (tabulation, carrying amounts, ...). Through these routes, the pupils living in a situation of discomfort have found a role and a place in the class group, taking active part in work, comparing their own proposals with those of their schoolfellows and drawing cues for reflection from others'proposals: this seems to be the most important aspect. The conquest of confidence in their own capabilities and the strong motivation towards school activities in general have surely represented the most important objective for cognitive development and, broadly speaking, for personal development.
Moreover, the didactic experimentation carried out in classes with socially-disadvantaged pupils has facilitated the formation of a different image of mathematics, as well as of an open and flexible mentality, critical but tolerant, whose results can surely be applied to didactics in general. The target is developing a type of didactics that does not lose sight of the aspects of curiosity, discovery and, ultimately, of the interest of pupils, safeguarding formal rigour at the same time.
According to some researches, verbalization is considered one of the fundamental instruments for achieving understanding and for developing reasoning capabilities, but it only represents one phase, which is often inadequate or not very concise, for the communication of what has been learned: the next step is the research and conquest of a symbolic system of communication. Therefore, it is necessary to pass to schematizations and then present symbols as an instrument for summing up and simplify writings.
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| Carla Caredda | Paola Vighi |
| Dipartimento di Matematica | Dipartimento di Matematica |
| Università | Università |
| Viale Merello, 92 | Via D'Azeglio, 85/A |
| 09123 Cagliari - Italy | 43100 Parma - Italy |