RICERCA ITALIANA     

Linguistic and representation aspects of the teaching and learning of mathematics

Università degli Studi di Genova

Abstract

Based on ongoing research within the 2003-2004 National Project with the same title and the current research results and perspectives at the international level, the new Project aims at producing new knowledge about the following research themes:
- linguistic aspects of the approach to algebra and to calculus;
- verbal representation of everyday life experience, elementary mathematical modelling and construction of mathematical concepts;
- epistemological and historical studies related to linguistic and representation aspects in the teaching and learning of mathematics;
- languages and communication in the mathematics classroom.
For each of the last two themes one Operative Unit will perform specific research, but the same themes will be considered by the other O. U.s too, in connection with the first two themes.
The first and the second theme will be dealt with by different operative units according to specific research tasks in a co-ordinate way.

The Project will take into account some important needs inherent in present Italian situation in the field of mathematics education:
- the need for increasing competencies in Mathematics Education in the Italian universities involved in the new system of pre-service teacher training, as a necessary scientific background for it;
- the need for producing and disseminating (even through the Internet) scientifically based materials for mathematics teachers' in-service training;
- the need for preparing a new generation of researchers in mathematics education.
The Project will try to pursue the intended scientific objectives with an eye to the listed needs.

As concerns research methodology, the Project refers to some peculiar aspects of the paradigm that is being developed in Italy since the end of the seventies:
- the characterisation as "research for innovation". In particular the theoretical framework of long-term studies is an object of research by itself, and it may undergo a significant evolution during the implementation of the research program;
- the role of teachers-researchers, who take part in the research activities of our operative units: teachers-researchers bring in the research team problems met by teachers in schools, evaluate whether the proposed solutions are meaningful for them, implement teaching experiments in the classroom being aware of the crucial inherent research aims;
- the nature of research in mathematics education: an interdisciplinary research based on tools and results drawing on different scientific domains (in particular: epistemology and history of mathematics, psychology, anthropology, sociology of education) as well as on its own tools and results; reference can be made to different theories, with the only requirement of local coherence (in the case of a specific research problem).

The Project will be carried out in two phases, lasting 15 months and 9 months respectively. During the first phase all Operative Units will perform most of the planned theoretical studies and set up and carry out the teaching experiments. During the second phase the analysis of these experiments, as well as the editing of papers and research reports (including those aimed at a wide dissemination in schools) will be undertaken. The second phase will be also characterised by the revision of the initially adopted theoretical framework, as well as by the formulation of new research questions.

Evaluation of the Project will be an ongoing process in both phases of the research. In the first phase there will be an internal evaluation through periodic meetings and the Project website. The presentation of partial results at international conferences will offer opportunities of an external evaluation. In the second phase there will mostly be an external evaluation by referees of international journals and scientific conferences: a concluding conference involving external experts will complete the evaluation process. <<<

Principal Investigator

Paolo BOERO Università degli Studi di GENOVA

Research Objectives

FOREWORD: In order to outline the complex aspects of this Project, as well as the variety of its objectives, it seems useful to sketch a picture of the situation in which the Project has been conceived and will be (hopefully) carried on.
University mathematics educators in Italy (like in other countries) have some important challenges and societal requests to meet:
- to reach or keep a sufficiently high scientific standard (in relationship with current status of mathematics education research in the world);
- to tackle relevant research problems related to the transformations of mathematics teaching within a rapidly changing society;
- to prepare future researchers and university teachers in mathematics education;
- to provide competencies for teachers training (it started five years ago for all school levels);
- to engage in local in-service teachers' training programs, as a way to improve teachers' professional knowledge, convey mathematics education research results and get large feedback for them.
We think that ALL THESE AIMS ARE INTERCONNECTED AND IMPORTANT (though for different reasons). In particular, the fourth and fifth aims need a good research background in the universities all over Italy (since most of the in-service teacher training courses are organised at a regional level). RESEARCH FUNDING is needed for the first three aims; at present a National Project like this (which involves seven universities in seven different regions) is the only accessible way to get research funds in Italy. Luckily some conditions are favourable in Italy in order to develop national research projects in the area of mathematics education:
- a tradition (starting from the end of the seventies) of COORDINATION of local (University) projects for innovation;
- a tradition of collaborative work in research, mainly through the National Seminar for Research in Mathematics Education (one or two sessions each year, since 1986);
- the productive experience of the first National Project (1999-2000), involving all Italian research teams. That Project allowed focusing on three main areas of investigation: difficulties in learning mathematics; mathematical knowledge construction and development toward the construction of theories; linguistic and representation aspects of teaching and learning mathematics. Some new constraints imposed by the Ministry on the dimension of national projects, brought to a consensual separation into three national projects (though keeping a high level of interaction and co-operation through the National Seminar and specific joint workshops).
THIS NEW PROJECT WILL DEVELOP (in 2005-2006) RESEARCH ACTIVITIES PARTLY BASED ON RESULTS ACHIEVED IN THE 2003-2004 PROJECT on the third general theme mentioned earlier. The area of investigation is the same - "linguistic and representation aspects of teaching and learning mathematics", as well as the O.U.s involved; some research questions have been re-formulated, or added, as an evolution of the research questions of the ongoing Project.

GENERAL OBJECTIVES
According to the outlined framework, this National Project is characterised by scientific objectives focusing on:
- the need to increase the quality of research in ALL research teams (in particular, the less experienced ones);
- the relevance of expected results as concerns their use to design materials or methodologies for teaching and teachers' preparation, and/or as tools for further research.

THEME 1: LINGUISTIC ASPECTS OF THE APPROACH TO ALGEBRA AND CALCULUS
The extension of mathematics instruction to larger and larger populations at different levels (high school, as well as university) has shown two critical points, where many students fail: the approach to algebraic language at the beginning of high school; and the approach to analysis at the end of high school/entrance to university.
As it will be discussed in the 2.2 Section, the answer consisting in postponing these difficulties in school revealed even worse than accepting failure. Mathematics education research elaborated very different hypotheses to INTERPRET and OVERCOME these difficulties.
The GENERAL OBJECTIVE concerning this theme will be to get further knowledge (mainly through experimental investigation) about the nature of the difficulties and the possibilities to overcome them according to different perspectives. In particular, one strand of research will try to identify and model epistemological and didactic obstacles; another strand will try to develop and model teaching sequences which integrate dynamically the use of verbal expressions (including metaphors), graphs and symbols, even with the help offered by existing software; while the third strand will try to connect performances at a higher level (e.g. algebraic level) to a revision/integration of the activities at the lower level (e.g. arithmetic). The fourth strand will try to develop practice and awareness (through the verbal channel) about the logic and linguistic features of the use of symbolic languages and mathematical registers of the natural language.

THEME 2: REPRESENTATION OF EVERYDAY LIFE EXPERIENCE, ELEMENTARY MATHEMATICAL MODELLING AND CONSTRUCTION OF MATHEMATICAL CONCEPTS
This theme is related to a traditional trend in mathematics education in Italy, concerning both innovation ("mathematics and reality" projects) and fundamental research (context-dependence of students' mathematical performances). Within our project, THE GENERAL OBJECTIVE will be to detect possible conflicts and synergies between students' conceptions as verbally (or graphically, or gesturally) represented, and elementary mathematical models, in the perspective of constructing robust "reference situations" (Vergnaud, 1990) for the development of mathematical concepts. Attention will be paid to cultural and educational problems posed by the rapidly increasing number of students in Italy coming from "other" cultures. Possible conflicts and synergies concerning representations of phenomena and related ways of reasoning inherent in the culture that students belong to will be considered.
In two O.U.s links will be made with the preceding theme.

THEME 3: EPISTEMOLOGICAL AND HISTORICAL STUDIES RELATED TO LINGUISTIC AND REPRESENTATION ASPECTS IN TEACHING AND LEARNING MATHEMATICS
As we will see in more details in Section 2.2., this theme is related to a one-century old tradition for mathematics education in Italy. The GENERAL OBJECTIVE of research will be to provide researchers with tools for cultural orientation of research, and teachers with tools for cultural orientation of teaching as well as methodologies to exploit suitable historical sources in the classroom. This means to improve knowledge about how languages and models enter mathematical activities, and what the history of languages of mathematics tells us about: difficulties met during the historical development of concepts , ways to overcome them; and simplification and other advantages brought by the invention of new languages.

THEME 4: LANGUAGES AND COMMUNICATION IN THE MATHEMATICS CLASSROOM
This theme is related to ongoing research at the national and international level concerning verbal and non-verbal communication in the mathematics classroom, and the role of different languages in students' mathematical activities.
It is related also to the need for developing students' use of natural language as a tool to think and communicate thought.
The GENERAL OBJECTIVE of research concerning this theme will be the analysis and modelling of experimental teaching situations aimed both at improving students' argumentative performances and constructing mathematical knowledge through verbal interactions.

Two O.U.s will deal with Theme 3, and two with Theme 4, among their major research subjects, while other O.U.s will make connections with them while dealing with other themes. <<<

First Results

FIRST PHASE: EXPECTED RESULTS
Expected results of the first phase will concern:
- at the very beginning (January/February 2005), re-thinking about previous research (performed within the 2003-2004 Project) and related new research questions;
- revision of the previous theoretical framework, in order to plan new experimental activities; in particular, revision of the methodology, taking the new research questions into account;
- new experimental activities aimed at collecting data related to the new research questions.
Most outcomes will be internal to the Project, with a major aim being the production of material for cross-comparison among Operative Units dealing with similar research questions (plans of teaching experiments, classroom materials, tentative analyses, etc. will be accessible on the Project website).
A number of outcomes will be external (research reports, contributions to meetings), specially aimed at comparing research perspectives and background results in national and international meetings: the PME-XXX Conference will offer a big opportunity in July, 2006 (see above).SECOND PHASE: EXPECTED RESULTS
Three kinds of scientific results are aimed at during and at the end of the second phase:
- results of a theoretical nature, which concern validating some research hypotheses about the considered themes, refining other hypotheses, formulating new research questions;
- validation of prototypes of didactic innovation and didactic materials, after being experimented by teachers-researchers in pilot classes;
- general design of materials for a wide dissemination of the didactical innovations and materials that have been tested with positive outcomes in the experimental classes of the teachers-researchers. However, the implementation and large-scale dissemination of materials is out of the purposes of the Project: applications for extra funding will be made to both local and national agencies, in order to support dissemination of results during year 2007, after the end of the Project (see above). <<<

Timescale

24 months

National and international background

SCIENTIFIC BACKGROUND

FOREWORD: In relation to the goals of the Project, by "scientific background" we mean the research performed and the results got both at the international level and by the Operative Units involved in the Project, that are useful to frame and tackle the research questions related to our programme. In the first part of this Section we will illustrate the title and present the research area, with some relevant references to international literature. In the second part we will sketch the scientific base related to the main themes of the research programme, while each operative unit in its B-model will be more detailed in showing specific previous results and references to current literature related to specific research questions. Information about the scientific base for general research methodology will close the Section.

THE TITLE OF THE PROJECT AND THE AREA OF INVESTIGATION:
Gathering in one title "languages" and "mental representations" can be justified by the narrow links that exist between mental representations and external representations in mathematical activities and learning of mathematics (as already highlighted by some authors, in Janvier, 1987; see Vergnaud, 1998 for a comprehensive development). A possible example related to such research in this project, is represented by the relationships between mental representations of properties of functions, and the representation of such properties by means of the system of symbols belonging to mathematical analysis.
The title refers to an important area of investigation in mathematics education at the international level in the last decades. With an eye to planned research activities, we can consider:
- the research perspective of mutual constitution of mathematical discourse and mathematical objects (see Sfard, 2000);
- Duval's research about the co-ordination of different linguistic registers meant as tools for thinking in individual problem solving, in the access to advanced mathematical knowledge, etc. (see Duval, 1995);
- the study of external representation problems in mathematical modelling activities (relationships between mathematical models and spontaneous mental representations, and their verbal or iconic representations, etc.: see Norman, 1993);
- the study of communication problems met in a mathematics lesson (in particular, the problems concerning the gaps between the language the teacher "must" convey as peculiar of mathematical communication and mathematical activities, and the language spoken and written by students) (for a general survey, see H Steinbring et al. (Eds.), 1998);
- Radford's research work about the approach to algebraic language through social interaction managed by the teacher (Radford, 2000)
- Balacheff's research work on "conceptions", according to a systemic perspective of knowledge where reference experiences, linguistic representations, concept properties and control processes work together (Balacheff, 1995)

THEME 1: LINGUISTIC ASPECTS OF THE APPROACH TO ALGEBRA AND CALCULUSThis research theme involves controversial questions about the ways of interpreting and overcoming some difficulties met by students, all over the world, while approaching elementary algebra and calculus.
The awareness of the controversial character of the research perspectives and our own previous results have led us to encompass differently oriented studies within our Project, in order to allow a true scientific debate to be developed and identify the potentials and pitfalls of the different approaches.
The main hypothesis of the Modena O.U., derived from previous research (see MALARA, 2002, 2003; MALARA and NAVARRA, 2001), concerns the possibility of preparing the transition from the arithmetic to the algebraic language, variables and functions, by developing suitable activities within arithmetic itself. For example, the approach to the function representing proportionality can be prepared by working on equivalent fractions. The hypothesis is related to the idea of a possible continuity between arithmetic thinking and algebraic thinking (see Carpenter & al, 2003; Carraher & al, 2000)The Genoa and Turin O.U.s' research on the approach to variables and functions is based on a hypothesis (derived from previous results: see Arzarello &C, 2001; BOERO, BAZZINI and GARUTI, 2001; BOERO, 2001; BAZZINI, 2002) regarding a co-ordination of different registers (verbal, symbolic, graphic) in activities concerning the elementary mathematical modelling of dynamic physical phenomena. This may create an experiential base for the dynamic aspects of functions (variables "covariance view", see Slavit, 1997 for a survey) and provide "grounding metaphors" (Lakoff& Nunez, 1997) to think in functional terms. In the same perspective of integration of different registers, the Pavia O.U.'s focus on CAS and their potential in relationship with a paper and pencil environment, is related to both international literature (see Hershkowitz and Kieran, 2001; Winslow, 2000) and their own previous results (see REGGIANI, 2000).
The Genoa and Rome O.U.s will deal with the approach to Calculus by studying those logic-linguistic features of conceptualisation and reasoning (especially related to proving in Calculus) where most students fail. Hairer and Vanner (1996) historical results, Vinner's research results on conceptualisation in Calculus and Harel's classification of students' proof schemes will be used as references to develop previous research work performed by the two O.U.s (see Ferrando, Proc. of CIEAEM-50) as concerns the relationships between the language used by teachers and students in high school approach to Calculus, and students' difficulties in university courses; and Bagni, in press,, who puts the evolution and progressive specialisation of verbal, symbolic and graphic registers into evidence in the historical evolution of concepts of function and limit).

THEME 2: VERBAL REPRESENTATION OF EVERYDAY LIFE EXPERIENCE, ELEMENTARY MATHEMATICAL MODELLING AND CONSTRUCTION OF MATHEMATICAL CONCEPTS
In this case the general perspective of the planned research is more coherent, and corresponds to the research trend concerning the role of context (meant as "task context", see Wedege, 1999, namely, as the experiential reference of tasks and classroom activities) in teaching and learning mathematics. In general, the research aims at studying the potentials (but also possible limitations and obstacles) inherent in different contexts in order to develop mathematical knowledge. Extensive theoretical and experimental work has already been carried out by the Genoa, Rome and Naples O.U.s: on the theoretical side as well as on the experimental side: see BOERO (2002); LANCIANO (1999); Guidoni(1999).
Various references to current mathematics education literature are relevant for our research work: see Freudenthal (1991), for theoretical framing of mathematisation processes; Gravemeier and Doorman (1999), as concerns the role of physical contexts for the construction of Calculus concepts; Hershkowitz, Dreyfus and Schwarz (2001), as concerns the problem of abstraction in the teaching of mathematics in contexts. Research will take into account not only studies supporting the idea of a big potential (for the learning of mathematics) related to suitable contexts, but also critical analyses about the limits of teaching and learning mathematics in contexts (in particular, the "transfer" problem and the ways to tackle difficulties in transfer: see Evans, 1999 for a survey).

THEME 3: EPISTEMOLOGICAL AND HISTORICAL STUDIES RELATED TO LINGUISTIC AND REPRESENTATION ASPECTS IN TEACHING AND LEARNING MATHEMATICS
This theme is related to a traditional attribute of mathematics education in Italy. Its origin can be recognised in the engagement of many Italian mathematicians at the end of XIX and beginning of XX Century in trying to develop a comprehensive perspective for mathematics education at all levels (from primary education to university courses), by rooting it both in the history of mathematics and in the contemporary debate about the nature of mathematics. Nowadays Italian research in mathematics education refers to this tradition with contributions that are recognised at the international level (see for instance the contributions by BAGNI, Bartolini, BOERO, Furinghetti, Grugnetti at the ICMI Study on History in Mathematics Education - see Fauvel & Van Maanen,Eds., 2000). We think that this kind of studies is still productive and important, especially in relationship with specific research themes and goals of this Project:
- identification of obstacles for classroom work and teacher training. About the nature of these obstacles, different positions exist in the international literature - Brousseau (1983) and Sierpinska (1985) stress the idea that some obstacles met in the history are inherent in the nature of mathematics, and this could explain why also students meet them in their approach to mathematical knowledge. Radford instead (2000) looks at the mechanisms of cultural transmission as responsible for obstacles. The O.U. of Palermo has already performed studies dealing with the of some obstacles met by students as "epistemological obstacles" (see SPAGNOLO, 1995)).
- construction of a repertoire of historical sources to be used in the classroom (for example, reference documents to situate some mathematical contents in a historical perspective, or "voices" to be imitated by students in an active way - see Radford, BOERO & Vasco, 2000). Concerning the latter use, the Genoa O. U. has developed an original didactical methodology (the "Voices and echoes game") to convey mathematical knowledge and forms of reasoning that are difficult to reach by students (see BOERO, Pedemonte & ROBOTTI, 1997).
- studies related to the epistemology of mathematical modelling and its educational implications at different levels, related to current historical and epistemological investigations about this subject: the results of DAPUETO&Parenti (1999) about the potential and limitation of the use of "real" contexts in the teaching and learning of mathematics refer to relevant contributions brought by Norman (1993); Blum&Niss (1991).
- studies about the linguistic aspects of the construction of axiomatic theories: with reference to classic studies of history and epistemology of mathematics (Hilbert, Pasch, Enriques, Veronese, Peano), the O. U. of Naples has developed a consistent background (based on historical considerations) concerning the variety of registers that can be used to organise mathematical knowledge in an axiomatic way in some geometry domains for classroom work. (see CASTAGNOLA & MORELLI, 2003)


THEME 4: LANGUAGES AND COMMUNICATION IN THE CLASSROOM
Under this title we have grouped a set of co-ordinated research activities dealing with bothproblems inherent in the linguistic aspects of individual mathematical performances (in particular, the difficulties related to the development of suitable forms of argumentation and the approach to mathematical registers of natural language); and the linguistic aspects of classroom social construction of mathematical knowledge, in peer as well as teacher-students communication.
The scientific background of these research activities draws on different theoretical frameworks, indeed one of the aims is to check the effectiveness of the different theoretical tools belonging to such theories (Vygotsky's and Activity theories; neopiagetian socio-constructivist theories; co-operative learning theories: see Lerman (Ed.) 1996; Cobb & Yackel, 1996; Bauersfeld, 1995; Sharan Y. & Sharan S, 1998).
The planned research will be carried out as a continuation of past Italian research in this field and will draw on the related results (in particular, with reference -respectively - to the management of classroom discussion, semiotic mediation and the teachers' role. to the development of argumentation for conceptualisation and approach to mathematical proof; and to cooperative activity in mathematical problem solving, see: Bartolini, Boni, Ferri &GARUTI, 1999; and BOERO, DOUEK &Ferrari, 2002. PESCI, 1998)

RESEARCH METHODOLOGY
Research methodology refers to the paradigm that is being developed in Italy since the end of the seventies (for a comprehensive presentation, see Arzarello and Bartolini Bussi, 1998). In our Project some peculiar aspects of the "Italian paradigm" are particularly relevant (as highlighted by all the Operative Units, with reference to specific methodological choices):
- first of all, the characterisation of "research for innovation", with relevant analogies and differences with other models of research(see Boero&Szendrei, 1997): we share with "research-action" theorists the aim of providing "useful" results for the renewal of teaching, but we think that "action" must be inspired, guided and modelled by "research" and that in several cases theoretical research does not need to have a direct follow-up in "actions" in the school. We share with theorists of didactical research as modelling of teaching and learning situations the need for reflecting on, and providing models for, what happens in classroom, but we think that this modelling purpose must be preferably pursued in a situation where teachers are involved in a progressive transformation of their teaching, due to their involvement in research. Obviously we share the need for a theoretical framework of experimental activities, but we think that during long term experimental studies the theoretical framework too must be an object of research, which may undergo an evolution;
- second, the role of teachers-researchers: they are engaged in our research teams as people who share the aims and many tasks with university researchers The important role of teachers-researchers is not only motivated by contingent reasons (to prepare competent people to support teacher training in pre-service apprenticeship), but also by substantial scientific reasons related to the "research for innovation" perspective (see MALARA & Zan, 2002;Bartolini &BAZZINI, 2003). In fact teachers-researchers bring inside the research team the problems met by teachers in the school, are able to evaluate whether the proposed solutions are meaningful for teachers, take part in classroom experiments being aware of the crucial inherent research questions;
- third, the nature of research in mathematics education: an interdisciplinary research based on tools and results derived from different scientific domains (in particular: epistemology and history of mathematics, psychology, anthropology, sociology of education) as well as on its own tools and results ( e.g.: the theoretical construct of the didactical contract, and more generally Brousseau's theory of didactical situations);
- fourth, the reference to a variety of theories, with the only requirement of "local" coherence (as far as a specific research problem is dealt with). <<<