1. Introduction
It is very difficult to synthesise the different points of view and the different approaches used by Italian researchers in didactic of mathematics in the study of the relations between technology and mathematical learning.
In the papers considered in this report the integration of ideas, methods and products of information technology in the mathematical curriculum (at primary, secondary and University level) is interpreted in many ways.
However there is a general agreement about two aims of this integration process: first, giving the students the basic ideas of computer science that are necessary in the information society, second using information technology ideas, methods and tools to improve mathematics teaching.
These two aims have been pursued by Italian researchers in their theoretical and experimental work using different methodological approaches; these approaches are supported by different background hypotheses about the role played by information technology in the process of mathematical learning or, more generally, in the general process of learning, where mathematics plays an important role.
The research and experimentation processes of integration of information technologies (at various school levels) has been neither linear nor homogeneous with respect to the technologies used, to the mathematical concepts involved and to the strategies of teaching approach.
However it is possible to emphasise some general trends.
2.1. Basic notions of computer science and programming activity in the mathematical curricula
The first use of information technology in the mathematics curricula is related to the introduction of fundamental notions of computer science and of programming activity.
This approach is motivated by two ideas: first it gives the possibility of going deeper into some important notions (such as variable, formal language, algorithm) used, sometimes with slightly different meanings, both in mathematics and in computer science; second it allows to emphasise important mathematical topics usually disregarded in the traditional curriculum, such as approximate calculations, algorithmic methods for solution of equations, integrals and so on.
The introduction of basic ideas of computer science and programming has been carried out using methods that are different but not opposite.
The papers of the first group can be defined of "research/action type" and focus on the choice of the notions of computer science to be introduced in the programming activities and on the method to be used in the integration between informatics and mathematics.
In (Reggiani, 1988) and (Reggiani, 1989) a proposal for a three year project (at level of lower secondary school) of introduction of programming activities (using BASIC) is presented, emphasising the goals and the contents of the activity. The project aims at developing some important ideas of computer science (such as: the computer as a "performer", explore its capabilities and its limitations, what are programming languages and data structures) and of mathematics (the meaning of variables in mathematics and in programming, the concept of algorithm, the construction of some elementary numerical algorithms).
A different project aiming at introducing programming activities in mathematics is described in (Iaderosa, 1991), (Malara and Guidi, 1988), (Pellegrino, 1989), (Pellegrino and Garuti, 1989 and 1990), (Pellegrino and Iaderosa, 1990), (Pellegrino and Malara, 1991). While the general aims are quite similar to the work of Reggiani, the main difference is in the use of LOGO, that allows a more structured approach to problem solving activities and to emphasise the recursion processes. Several situation/problems faced by the students and the methodology for class activity planning are presented; (Malara and Garuti, 1991) and (Pellegrino, 1992) are very significant of the activity of this group of researcher. In the first a teaching experimentation is described, where students face the problem of implementing algorithms on the computer; with a careful management by the teacher the use of a table for the organisation of problem data is an useful tool for an analysis of the procedure, for the development of new procedures, and for the start of a consideration on his own and other people mental processes.
In (Simonetti, 1994) and (Bondi, 1994), two experiences are described, where LOGO and in particular the graphic turtle are used; the aim is to develop skills about arithmetic and algebraic properties (through their visualisation on the screen) and to help discovery of properties in plane geometry and in transformation geometry.
The project I.M.P.A.R.A. (Informatica: Metodologia Pluridisciplinare in Attivita` di Recupero e Approfondimento) described in (Alberti et al., 1994a and 1994b) and in (Borromeo et al., 1994) aims at integrating programming activities and the use of computers as a tool in lower secondary school not only in Mathematics but in all subjects: in languages learning, arts (music and drawing) and in technical education. Without changing the traditional curriculum, computer in used in improving teaching effectiveness and the student logical and creative development; in the activities more closely related to Mathematics, BASIC and LOGO and the first elements of PASCAL are introduced.
Forcheri et al. (1990), Bottino and Furinghetti (1990), Bottino and Furinghetti (1992) and Bottino et al. (1993) describe an experimentation in the first two years of upper secondary school for the introduction of elements of computer science in the mathematical curriculum. This project began to be developed before the National Plan for Informatics (PNI) organised by Italian Ministry of Education and then it has taken into account also the suggestions worked out by this plan. Several problems related with the integration of basic concepts of computer science and mathematics are discussed: the importance of this new subject in a curriculum devoted mainly to basic education, the distinction between basic and more advanced notions, the balance between theory and practice, the relations between informatics and mathematics, their common aims, the problem of evaluation. Other papers by the same authors analyse the difficulties and the obstacles of cognitive, epistemological and didactic nature that arise in this activity.
Arzarello et al. (1993) analyses the programming learning as a cognitive apprenticeship, that is, as an apprenticeship where the interaction between expert and novice is aimed at enhancing the cognitive and metacognitive skills of the latter. The authors describe the intellectual abilities involved in learning programming and emerging cognitive conflicts by the novices. Moreover they put in evidence the importance to develop intermediate supports in order to help the transition from one problem solving environment to another. These intermediate supports are conceived as tool to stimulate a flexible and rich anticipatory thought, which seems to be necessary both for the mastery of the mentioned intellectual abilities and for recognising and overcoming the cognitive conflict.
In (Reggiani 1991, 1992 and 1994) and in (Reggiani and Vercesi, 1993) the analysis of the results of teaching experimentation focuses on learning processes and on difficulties connected with the concept of variable and with the fundamental notions of programming: the authors describe the problems that student face in the use of input instructions, in the use of a variable as a counter or as an accumulator in a loop, in the use of variables with index. These difficulties can be related to the logical structure considered or can be of formal and operational type, related with the specific programming language used.
In (Chiappini et al. 1990 and 1994) and in (Pellegrino and Malara, 1991) several difficulties arising in problem solving with computer are analysed. These difficulties are due to: i) language deficiencies (for communicating and representing solution strategies and using artificial language coding); ii) planning deficiencies connected with the internal management of space-time image by the mind, associated with the evolution in time of the resolutions according to a particular working hypothesis; iii) cultural deficiencies regarding the mastery of the relationship between possible reasoning by man, and forms of reasoning incorporated, and/or that may be incorporated, in the computer. Some examples of teaching experiments aiming at overcoming these difficulties are presented and discussed.
In (Fasano 1992 and 1994) the aspect of an epistemological analyses about contents of mathematics, computer science and technology for the new curricula are described. The goal is the building of concept maps in computer science both to discover and apply the link at a technological level as well, with the mathematical concepts we find in the Italian new curricula in elementary and middle school
2.2. The computer as a tool for learning mathematics
Using computer as a tool is a different way of introducing information technology in a mathematics curriculum.
In this approach the student is helped in learning mathematics by the use of flexible software packages such as spreadsheets, symbolic manipulators or special packages devoted to a particular field; many of these software tools (such as spreadsheets and manipulators) are widely used in the society, for purposes different from teaching.
We examine some papers describing the research activity related with this approach, considering the different school levels in which the experimentation is carried out.
At lower secondary school level we have two groups of papers.
In the works by Arpinati and Barozzi the computer is considered as a mathematics laboratory i.e. as "a tool whose capabilities and limitations allow the teacher to reach some didactic aims". The computer is used for its computing capabilities and for its limitations that arise when numbers too big or too small are considered. In (Arpinati, 1988 and 1990a) the aim is to discover and understand the structure of machine numbers. In (Arpinati and Barozzi, 199) with the help of the teacher, students can see, though some numerical experiments, how the set of machine numbers (a finite and discrete set) is too poor and not adequate when it is compared with the sets of integer, rational and real numbers, that students are learning in their mathematical studies.
In (Arpinati, 1990b) and (Arpinati and Barozzi, 1992) with the help of the teacher and with an essential use of the package Mathematica some classical mathematical topics are revisited with computer experimentation: factorisation and prime numbers, divisibility criteria, numerical sequences and the structure of the irrational number pi.
In (Dettori et al., to appear) and (Malara et al. , 1992) the approach to algebra by means of a spreadsheet is analysed; the latter work describes the possible use of a spreadsheet in a mathematical curriculum for students aged from 11 to 16; the general features of the package are described and several examples are presented. Among them, the studies of regularity in numerical sequences, the discovery of proportionality laws, statistical tables.
Dettori et al. (to appear) emphasises the algebraic notions that can be developed by students when they work with a spreadsheet; several situation/problems are presented and the solutions worked out in a purely algebraic setting and in the spreadsheet based setting are compared, noticing capabilities and limitations offered by the computer based environment.
Fasano (1993) describes the goal and the results of the research about the building of a tool, in this case by a hypercard, on a mathematical subject by student aged 11-12 years. The hypothesis is that this activity can develop cognitive mathematical frames in boys and girls. In this paper concept maps are described as : bridge activities between the organisation of pupils' knowledge and the tool data; a means of verification for the teacher. Fasano (1994) describes the result of research aimed to the development of logical-mathematical thinking by the use of an AI software. The paper also shows how the use of some video-recorded activities has helped the teachers to choose more appropriate didactic strategies.
At level of upper secondary school Bottino et al. (1994), Furinghetti (1992), Furinghetti and Paola (to appear), according with the guidelines of "Piano Nazionale per l'Informatica" (PNI) promoted by the Ministry of Education, study the different possibilities in using computer as a supporting tool in the mathematics teaching activity in order to construct appropriate mathematical concepts and to overcome the traditional idea of mathematical activity as a rigid application of facts, rules, formulas, procedures suggested by somebody else (the teacher, the textbook,...)
Only a few teachers use pocket calculators in the classroom in Italy; the experience described in (Castagnola, 1994) shows how a careful use of a pocket calculator with graphic capabilities can help facing non trivial problems (finding roots of equations, graphing functions, conjecturing limits, solving ODE's).
At University level, Capuzzo Dolcetta et al. (1988a and 1998b) describe an experimentation developed at University of Rome of introduction of computer as a visualisation tool in a Calculus course; computer is used in the classroom by the teacher and in the laboratory by the students.
Also Boieri (1988) describes the use of computer in a Calculus course (Analisi Matematica II); the package used by all students in the laboratory is the symbolic manipulator MuMath, while a selected group of students write some programs using the challenging language Musimp.
Elia et al (1993), Galizia and Malaguzzi (1992), Galizia and Mascarello (1993b, 1994a, 1994b), Galizia et al. (1993), Sargenti (1992) use different software packages (mainly Derive and software for graphical visualisation) as a tool for teaching and learning integration theory, Fourier series and Fourier transforms. They focus their attention in the construction and experimentation of a curriculum where these topics are studied both at the level of upper technical secondary school and at the level of Engineering faculty, emphasising the continuity of the learning process and the relation with the application to Electronics and Information theory. In (Galizia and Mascarello, 1993a) capabilities and limitations of different software packages in the study of integration and Fourier series and transforms are analysed; in (Elia et al., 1993) the connections between mathematics curriculum and the needs of Information theory teaching are studied.
2.3. Use of microworld based systems for the development of mathematical abilities
A third way of using information technology in mathematics curricula is based on microworld systems.
There is a general agreement on the effectiveness of the use of microworld based systems in mathematics learning: the learner is encouraged to explore the environment that is accessible through the interface and that embodies a model of a domain of mathematical knowledge.
The exploration of a microworld, even with different features depending on the systems, can help to achieve two objectives: the learner improves his/her solving strategies to solve the task; he/she can build meanings that allow to set the relation between mathematical objects and their representations.
Some researchers studied the conditions under which a microworld can allow to achieve these important aims.
Bottino et al. (1994) and Bottino and Chiappini (1995) study the opportunities of learning in the field of algebraic problem solving through ARI-LAB, a microworld based system, planned and developed by them.
In ARI-LAB the opportunities supplied by hypermedia allow the users to build their own solutions to arithmetic problems by navigating through different computer-based environments that have been integrated in the system; the environments include microworlds, databases containing sets of solved problems and a communication environment. The combination of hypermedia with communication facilities allows pupils to cooperate in order to solve problems, or even to interact with the teacher, through a local or remote network.
In these works technical and pedagogical aspects are closely connected. On one hand the authors base their work on the research on hypermedia and communication systems and on the design and implementation of visual microworlds. On the other hand they take into account the research in mathematics education with particular reference to the studies on applied problem solving and on the role of visual representations as educational mediators and interactive learning.
The evaluation of the system ARI-LAB has been worked out through many experiments involving elementary, lower secondary school and disabled students (see the last paragraph).
Another important microworld is Cabri-géomètre, that has been introduced quite late in Italy and experimented only in the last years (see Boieri, 1994 and Pellegrino, 1994 and the two reports about geometry in this volume). The first experiments gave rise to a very high interest, as shown by the development and the wide circulation reached by the bulletin CabrIRRSAE (published by Irrsae Emilia Romagna).
Researchers and teachers focus their attention mainly on the use of Cabri-géomètre in teaching geometry, in the preparation of learning units, in the search for an optimal way of using the possibilities offered by this software; the list of related papers is too long and can not be a part of this general report: the interested reader can see CabrIRRSAE.
Balzamo et al. (1994) describes the theoretical background underlying the realisation of GeT (Geometria e Trasformazioni) and describes the main features of this system, emphasising its structure, its capabilities and the learning strategies that can be supported by its use.
With GeT it is possible to work with geometric transformations following three different approaches: dynamic, geometrical and algebraic.
In the dynamic approach (environment "Movimento") the transformation is the result of a movement worked out by a mouse displacement; the geometrical approach (environment "Costruzione") describes the transformation as correspondence between pairs of points, while the algebraic approach (environment "Equazioni") emphasises the algebraic relations between the initial and the final point in a transformation.
In GeT these three points of view are at the same level; it is possible to switch quickly from one to another, setting relationships between them and noticing the differences between them.
Another approach to the study of geometric transformations through the use of computers is described in (Avanzini et al., 1988).
2.4. The use of technology in the transmission and diffusion of mathematical contents
A fourth possible use of information technology in a mathematical curriculum is based on the use of tools and techniques for the communication and the diffusion of mathematical contents and concepts; the possibilities offered by technology (films, videodisks, hypertexts) are used in order to make mathematics more accessible to be taught and/or to present it to wider audiences. The products, which have been developed according to this philosophy, can be used in a more flexible way by the learner for individual and self-paced learning.
An interesting wide scale experimentation of continuing education and distance learning has been carried out by Consorzio Nettuno, that prepared a course in Ingegneria Automatica and Informatica (later followed by other courses) at intermediate University level. This program involves five mathematical courses; the lessons are available on video cassette and are broadcast during the night by the public television network.
Many works by Emmer are based on the use of the video technology; among them we find videotaped lessons on mathematics subjects (Emmer et al., 1989, Emmer and Loreto, 1989), where a large use of computer animation and simulation is made and short movies, addressed to a wider audience, that explore interesting connections between mathematics, fine arts and history (Emmer, 1990b, 1994, Emmer et al., 1990). In the other works the same author studies the role of different media (computer, film and also math exhibitions) in the learning and teaching of mathematics "in and outside mathematics departments".
Also Speranza, 1989 carries out a critical analysis on the use of films in the learning of mathematics, emphasising the change from a way of teaching based mainly on the oral and written communication to a way of teaching where images and animation play the main role.
In (Alloatti et al., 1994a, 1994b), (Giannetti and Scarafiotti, 1993, 1994), (Giannetti et al., 1993a, 1993b) the project, the development and the experimentation of a hypermedia system, called HyperMath, are presented; HyperMath deals with Calculus at University level and can be used also for distance learning.
These works describe and analyse the whole system and several particular aspects: the main features of the user interface, the organisation of knowledge, the navigation through it and the interactivity, the concept of psychology of learning embodied in the system, the evaluation and the self-evaluation aspects.
Another experimentation carried out as a part of Progetto Strategico of CNR "Tecnologie e Innovazioni Didattiche" deals with the production and the use of interactive videodisks in the teaching of set theory. The features of this system and the results of its experimentation are described in Bonotto and Zanardo, 1989.
The information technology is used also to provide teachers with tools for bibliographical research (Lucchini and Ferro, 1989 and Lucchini, 1992).
2.5. Teacher training for integration of information technology in mathematics curricula
The different ways to develop an integration between mathematical curricula and information technology described in the previous paragraphs arise this question: what conditions can allow spreading this experimentation work, taking into account the teachers opinions about this process and the problems connected with teachers training.
Bottino and Furinghetti (1996) analyses the teachers conceptions about informatics and its teaching in connection with mathematics. They focus on the teacher's role by analysing the different choices taken by mathematics teachers when faced with a curriculum reform contemplating the introduction of informatics in secondary school courses. The fundamental assumption is that these choices are the consequence of the conception teachers have elaborated on their own about informatics in relation to the teaching of mathematics. The work is based on a detailed analysis of some case-studies and, on the base of this analysis, a typology of teachers' conceptions is described.
The problem of teachers training is considered in (Chiappini et al., 1990) and in (Dapueto, 1993); the authors describe two experiences of planning and developing training courses for in service teachers of lower and upper secondary school. The latter work analyses some problems connected with the introduction of computers and innovation of mathematical curricula: the integration between computer and mathematics (mathematics with the computer, mathematics for using computer, mathematics for knowing computer); teachers' difficulties in facing up the changes and connecting "knowledge" and "scholastic knowledge".
The problem of teachers' training in the context of PNI is considered also in (Bottino and Furinghetti, 1991) and in (Forcheri et al., 1991); the authors analyse if the PNI training is effective in meeting both the teacher's expectations and the need for developing correct directions in teaching.
2.6. Mathematical learning, technology and disabilities
We can conclude this review of the research and experimentation about the connections between information technology and teaching/learning of mathematics describing some interesting experiences in using computer to improve school integration of disabled subjects.
Experimentation in this field is considered with great attention by teachers, parents and social service workers whose aim is to insert disabled pupils in the school system, according to the laws on this subject.
Contardi et al., 1995a and 1995b describe the features of some programs ("The supermarket", "The line of numbers", "Indiana Jeans"), written by the authors for students with mental deficiency; the aim is to help them in the construction of mathematical competencies and abilities (classifications, relations, mappings, numbers and operations, displacement in the plane and Cartesian co-ordinates) and to improve their autonomy also in mathematics.
(Chiappini and Mantovani, 1994) is devoted to an experimentation of using ARI-LAB in a class with some disabled students in order to improve their capabilities in arithmetic problem solving.
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| Paolo Boieri | Giampaolo Chiappini | Margherita Fasano |
| Politecnico di Torino | Istituto per la Matematica | Università della Basilicata |
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