1. Introduction
In this chapter, we shall present and review some of the studies that have been published by Italian mathematics educators between 1988 and 1995 and that are explicitly focused on geometrical reasoning in the classroom activities. The importance of classroom experiments in the whole of Italian research in mathematics education could hardly be overestimated, because of tradition on the one hand (see for instance the reference to the seminal work of Emma Castelnuovo and Lucio Lombardo Radice in the first part of Barra et al.[1], 1992) and of strong social pressure on the other (see the reference to the deep changes of the last decade in school programs after a very long period of wait-and-see policy). The search for innovative ways of teaching geometry has produced many studies with different focuses. Their diffusion abroad has been different, as it appears from the location of the papers in the enclosed list. The need of offering professional tools to a larger and larger number of teachers has produced exemplary reports of successful school experiments, with general methodological advice and related training workshops in national meetings for teachers.
The importance of the national diffusion of these materials cannot be underestimated, as it has taken the place of still lacking institutional programs for pre-service and in-service teacher training. The need of reproducing the experiments effectively has produced different trends of research, that we can summarise as follows:
2. The need of offering professional tools to teachers of geometry
A great effort has been made by the Italian mathematics educators to offer didactic materials to teachers. For instance, we can refer to the systematic work of the groups in Rome for in service teacher training in primary school (Bernardi et al. 1990-91) and of the group in Florence for in service teacher training in secondary school (Bondi 1994, Campedelli 1992, Campedelli 1995, Fabbrizzi 1994, Simonetti 1994). Several topics are covered, such as paper folding, plane tiling to realise artistic periodic design, elementary methods for the computation of [[pi]]. The concept of measure is discussed in details in Iacomella and Marchini (1990), who address primary school teachers.
The introduction of the computer in the classroom has been gradually taken into account. LOGO (Fabbrizzi 1994) is used as a tool to discover the properties of figures and geometric transformations in grades 6-8. Cabri Géomètre is more and more considered in order to emphasise the formative value of Euclidean geometry, that has been underestimated in the algebraic approach. Pellegrino (1995) for instance proposes the solution of a rather difficult problem of plane geometry by means of Cabri and discusses the didactic potentialities of this kind of solution. The same author (Pellegrino and Zagabrio, in press) develops a comprehensive approach to geometry through Cabri-Géomètre: interesting topics are covered, such as the properties of circular inversion and the concepts related to pencils of circles.
An original trend of research is represented by Barra, who explores the relationships between three-dimensional space and n-dimensional spaces, suggesting activities to develop intuition through analogy and induction (Barra 1994); in other papers the same author examines the possibility of developing proofs without words by linking geometry to other mathematical fields (Barra 1990b, 1990c, 1995a, 1995b, 1995c).
The national meetings for teachers, when hundreds of motivated teachers are present, have been the right place to popularise research results. Hence we see several papers, published in proceedings of national conferences, aiming to link research results and school practice (e.g. Barra 1995, Gallo 1994b, Iaderosa and Malara 1995, Mariotti, 1994c, 1995a).
3. Analysis of effective teaching experiments
The complex relationship between research results and school practice is directly approached by Italian researchers in many articles.
A noticeable number of research studies papers provide a detailed description of teaching experiments in geometry at different school levels; these descriptions deal with the design, the functioning and the products of the experiment. The major aim is to analyse and discuss the consequences of teacher's behaviour on classroom management.
For instance difficulties and misconceptions about angles in 3D geometry are considered in a questionnaire, set up by Longo Bruno (1995) for secondary school level, whereas the problematic aspects of the intersection of geometrical figures is dealt in Medici and Vighi (1991).
Experiments in innovation have been done and reported for all the school levels. The description of classroom management is focused with emphasis on the strict interlacing between contents and methodology, while the interest in pupils' behaviour is shown by the detailed report of solution processes in classroom activities. Different approaches are considered.
The Pavia group (Ferrari and Sforzini 1995) emphasises mathematics and creativity, by means of constructions with ruler and compasses, that are explicitly linked with artistic work by means of string design and coloured tiling.
Manipulative materials, such as tangram (Pellegrino and Iaderosa 1990), are used to modify student passive attitude into creativity in mathematics activity at grades 6-8. The detachment from manipulative towards combinatorial reasoning is fostered by means of a long term project centred on the problem of unfolding polyhedra (Arpinati Barozzi and Pellegrino 1991, Pellegrino and Arpinati Barozzi 1996).
Problem posing and hypothetical reasoning in the geometrical realm is analysed in Malara and Gherpelli (1994). Pupils are led to pose problems on elementary geometrical plane figures, through the formulation of their texts, in small group situation.
The basic idea of invariant in geometrical transformations is focused on in Malara (1994) and Pincella and Malara (1995). Starting from the analysis of Italian school programs and tradition, the authors propose an innovative approach centered on the idea of invariant; pupils are introduced to experimental activities and guided to control and systematise their observations by means of a two-entry table, that highlights the correlation among different parameters involved. In the same stream of research, the possibility of the ideal extension of geometrical transformations to the whole plane is explored through the systematic study of computer screen figures; difficulties and failures are clearly analysed and related to the existing literature on geometrical reasoning (Malara 1995).
According to the tradition of Emma Castelnuovo, the presence of manipulative and, more generally, of experimental activities seems to be a distinctive feature of most of the Italian experiments in innovation in teaching geometry. The school programs themselves highlight the approach to geometry as the early representation of physical world. In this stream we have several examples of systematic exploration of some real life or cultural fields of experience, where problematic situations can be modelled and solved by means of the systematic recourse to geometrical concepts and procedures.
Architecture and town-planning are considered as source of interesting geometrical problems in Barra (1990a). Cartography is suggested for geometrical activity in the classroom in Barra (in press). The urban space, as an example of macro space[2], is approached at elementary school level in a holistic way by Ardizzone and al. (in press), with the aim of improving children's attitude towards the learning of geometry, using their bodies and senses too. In order to overcome some widespread difficulties in space orienting, the country space is approached by Grugnetti and Uselli (in press) with 6th graders, up to the drawing of a map. The mega space of astronomy is focused on by Lanciano (1994) at the elementary school level; the author aims to study the evolution from the initial explanatory models to concepts through experiments, observations and conceptual perturbations. The field of experience of sun shadows is presented and analysed from a cultural perspective by Rondini (1989). The field of experience of representation of visible world by means of perspective drawing is approached by Bartolini Bussi (in press a), Costa and al. (in press), Ferri (1993). The field of experience of mathematical machines (see also the report of Marchi et al., in this volume), is studied by Bartolini Bussi, Pergola and Zanoli. Within the Italian research on innovation, the variety of contexts for student activity may well be assumed as a common feature for geometrical activity.
4. The study of mental processes in geometrical reasoning
Few important research projects have been carried out in the past 10 years, with the specific objective of characterising different aspects of geometrical reasoning.
In this section we review research papers focusing on the mental processes involved in geometrical reasoning; progressive attempts can be found aiming to understand what and how students learn from teaching experiments and to construct models for mental processes in geometrical reasoning.
The theme of mental images inspires the teaching experiment about space geometry in the elementary school of Azzali Carminati and Visintin Macino (1993): the theoretical results of general psychology are implemented in classroom experiments in 1st and 2nd grades. Mental images are introduced to analyse proofs without words, when different mathematical fields, such as calculus and number theory are explicitly related to geometry in Barra (1995c).
A wide and deep analysis is devoted by Gallo to specific problem solving situations in plane geometry at the upper secondary level. The main point (Gallo and Testa 1991) concerns the link between a verbal description and a graphic representation of a geometrical figure; the experimental design concerned 9th grade pupils and was based on a situation of communication in which the task alternatively required the coding and the decoding of a verbal message describing the drawing of a geometrical figure. The main hypothesis concerns the interaction among task, model and action, involved in the formation of specific mental representations. In problem resolution situations pupils recall, use and construct models; as far as the coding task is concerned, the results highlighted the role of standard models of geometrical figures and stereotypes of language expressions: different types of conflicts arise between figures and their mental representations, which the author classifies as intrafigural and interfigural.
In the decoding task, main difficulties appear when the co-ordination between the text and the action is required. Besides the characterisation of models, an interesting aspect was highlighted. During the solution process, the mental model plays a specific role of control; the model organises the available information and guides the subject's cognitive activity in a process of descending (from the model to the action) and ascending control. The analysis is carried out according to both the degree of structuring and the degree of applicability of the model; the variety of behaviours is described and analysed, whereas the implication of different types of classroom situation (individual versus pairs interaction) are discussed (Gallo 1994a, 1995).
At a different level of analysis pertains the research project carried out by Mariotti in the reference frame of the theory of figural concepts[3]. The main thesis is that geometry (in elementary Euclidean terms) deals with particular mental objects, the figural concepts, which are characterised by the fusion of a figural component and a conceptual component, so that geometrical reasoning results into the dialectic interaction between these two components. In order to deepen the analysis of the interaction between the figural and the conceptual components, an experimental design was carried out aiming to analyse the mental processes involved in geometrical reasoning (Mariotti 1992a, 1992b, 1995c). The analysis, based on the protocols of individual interviews, showed how difficulties and errors can be interpreted in terms of a rupture between the two components, whereas successful reasoning can be interpreted as a good harmony between them. The study concerned three different age levels (the end of the primary school 10 up to 11 year old / the end of the middle school 13 up to 14 year old / the end of the High school 17 up to 18 year old); interesting aspects emerge from the analysis of pupils performances in the solution of simple problems on folding/unfolding polyhedra. In particular (Mariotti, 1991), age invariant elements are highlighted (such as the influence of the figural aspect, the influence of standard images) and a classification of the solution process is attempted according to different levels of contribution of the conceptual aspect (Mariotti, 1993a, 1993b). The results obtained inspired, a teaching experiment, aiming to compare the interview situation and the classroom situation. Interesting results (Mariotti 1994a, 1994b) came out of the comparison showing the role of social interaction in triggering the dialectics between the figural and the conceptual aspects.
The problem of drawing often appears in relation to the mental processes involved in geometrical reasoning (see for instance its heuristic role in Scali in press, Tizzani and Parenti in press). A specific approach to the problem of drawing, as far as its potentialities in concept formation and geometrical problem solving is contained in Bondesan and Ferrari (1991) and Ferrari (1992). It is argued that the drawing is crucial in the search of a strategy in arithmetic and geometrical problems as well. The analysis carried out suggests that the perception of a drawing as an object autonomous from the graphic construction performed, is achieved after a difficult and contradictory process. The functional role of drawing in the development of children, that is assumed in the whole of these research studies, clearly contrasts the diagnostic role that is given to drawing in the piagetian tradition literature. We shall come to this point later. The functional role of drawing is considered in a different perspective in the Cabri environment (Ferrari, in press; Mariotti, in press).
5. Towards a comprehensive theoretical framework
In this section we review papers with progressive attempts to define a theoretical framework for innovation in geometry, to detect the fundamental elements of the teaching and learning processes in the classroom.
We have already highlighted that fields of experience from everyday or cultural life are systematically used by many Italian mathematics educators. A considerable effort of systematising this trend has been made by Boero (Bartolini Bussi and Boero, in press), by means of the analysis of the relationship between teaching learning geometry and fields of experience. The analysis is done on the basis of two complementary standpoints. According to the first standpoint, the focus is on the evolution of student internal context (i.e. the choice of tools in geometrical problem solving, conceptions and so on); according to the second standpoint, the focus is on the quality of geometrical activity, i. e. the culturally and epistemologically relevant aspects of the historical evolution of geometry.
We may quote research papers related to each of them.
As far as the former is concerned, Tizzani and Parenti (in press) examine the drawing activity in different fields of experience, that refer to spaces of different sizes (micro, meso and macro-space), by eliciting student mental dynamics. Scali (in press) examines two different functions of drawing in elementary geometrical modelling of astronomical phenomena, i. e. drawing as external representation of an hypothesis or a piece of knowledge and drawing as a heuristic tool, allowing to generate new hypotheses or take into account new possibilities. Careful analyses of student protocols are provided.
As far as the latter is concerned, Costa et al. (in press), analyse the function of perspective drawing in a long term teaching experiment; they consider very young learners (3-6 graders) and their early productions of 'proofs' to justify an astonishing result in perspective drawing produced by arguments and not by experiments. Boero and Garuti (1994) report on a long term teaching experiment (6-7 grades) in the field of sun shadows, concerning the production of geometry statements and the comparison between the statements produced and the statements contained in the textbooks.
The systematic research on the fields of experience aims, in the long run, to give teachers criteria to choose among different fields of experience and to accomplish their choice in a rational way. The experimental research studies in the classroom aim to detect the limits and the potentialities of activities related to different classroom contexts.
A specific related trend of research (for a discussion of this issue, see Bartolini Bussi, in press c) concerns the analysis of classroom processes in the activity of teaching - learning geometry. Different issues are addressed, such as the analysis of the role of the teacher with reference to some relevant aspects of geometrical activity. The research studies in this trend share a common reference to a specific model of learning, or, better of the teaching - learning process, i. e. the role of the teacher as a guide in the process. The reference to Vygotskij is usually made explicit by the researchers, with explicit borrowing of theoretical constructs from activity theory.
For instance, Mariotti (1994b e 1995b) focuses the role of the teacher in the construction of a definition as a fundamental element of the classroom culture. The theory of figural concepts in this case functions as a tool to design the experiment and to analyse the resulting processes.
When the classroom dynamics are in the foreground, different tools are considered. Bartolini Bussi (1992) studies a long term teaching experiment on graphs in the Cartesian plane (5th grade) by means of the Vygotskian construct of internalisation. Bartolini Bussi (in press a) studies the process of internalisation of strategies in real life drawing, within a long term teaching experiment (1st and 2nd grades), based on the systematic alternation of individual tasks and collective discussions orchestrated by the teacher. A methodology of analysis of collective discussions, based on Vygotskij and Leont'ev seminal work is presented and applied to a case of discussion on the geometrical concept of point of view (Bartolini Bussi and Boni, 1995). By means of Leont'ev's activity theory and of Vygotskij's construct of semiotic mediation Bartolini Bussi considers the functional role of drawing in the overall development of the child on the one side and the phenomenological role of perspective drawing in the genesis of modern geometry. A historical approach shapes the project and is made explicit to pupils, too, in spite of their young age (Ferri, 1993). Two different readings are offered of a long term teaching experiment on perspective drawing that has been tried out from grade 2 to grade 5 (Bartolini Bussi, in press b), to show that semiotic mediation by cultural artefacts must be strictly interlaced with specific and intentional experience of social interaction in the classroom. In this case theoretical analysis of Vygotskian approach is linked with the analysis of classroom experiments.
6. Concluding remarks
The complex of research studies that has been reviewed in this report is multifaceted and witnesses the richness of the different contributions; behind the differences there is a strong common tension towards the problem of innovation in the classroom. Teacher training in geometry is focused with strong efforts in popularisation. Specific experiments and research studies produce data and suggest research questions that are integrated into the progressive systematisation of theoretical frameworks; the theoretical foundations produce the conditions for designing specific teaching experiments and for interpreting the classroom processes; results coming out of teaching experiments contribute to the enrichment of the theoretical frameworks in a dialectic synergy between theory and practice.
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| Maria G. Bartolini Bussi | Maria Alessandra Mariotti |
| Dipartimento di Matematica Pura ed Applicata | Dipartimento di Matematica |
| Via Campi 213/B | Via Buonarroti 2 |
| I - 41100 Modena | I - 56127 Pisa |
| E-mail: BARTOLINI@DIPMAT.UNIMO.IT | E-mail: MARIOTTI@DM.UNIPI.IT |