Recent Italian Research in Mathematics Education

Seminario Nazionale di Ricerca
in Didattica della Matematica
In memory of
Francesco Speranza
Recent Italian Research in
Mathematics Education
edited by
nicolina a. malara, pier luigi ferrari
luciana bazzini, giampaolo chiappini
ISBN: 88-900029-3-1
The book was realized with the financial support by the Department of
Mathematics of Modena and Reggio Emilia University.
A copy of this book can be requested to Nicolina A. Malara, Dipartimento di
Matematica via Campi 213/B, 41100 Modena, Italy. e-mail <[email protected]>.
It can also be found in the web site of Modena and Reggio Emilia University
<http://www.matematica.unimo.it/matheduc/home.htm>
Index
Introduction
part 1
Ferdinando Arzarello
Present Trends of the Research for Innovation in Italy:
a Theoretical Framework …….……………………………………
Nicolina A. Malara
p. 9
The “Seminario Nazionale”: an Environment for Enhancing
and Refining the Italian Research in Mathematics Education ……..
p. 31
Nicolina A. Malara
Francesco Speranza as a Mathematics Educator:
Values and Cultural Choices ………………………….…………...
p. 66
Carlo Marchini
The Philosophy of Mathematics according to
Francesco Speranza ………………………………………………...
p. 83
PART 2
A Survey of the Italian Present Research in Mathematics Education
Keys for Classification of Abstracts ………………………………
p 97
Abstracts of Selected Papers ………………………………………
p 99
Addresses of the Authors’ papers …………………………………
p. 181
INTRODUCTION
This volume presents an overview of the Italian Research in Mathematics
Education, as it has been developed in recent years.
It has been prepared on the occasion of the International Conference ICME9
(Tokyo, 2000) and constitues an ideal continuation of the books “The Italian
Research in Mathematics Education: common roots and present trends” (eds.
Barra M., Ferrari M, Furinghetti F., Malara N.A., Speranza F., CNR, 1992) and
“ Italian Research in Mathematics Education: 1988-1996” (eds. Malara N.A.,
Menghini M., Reggiani M., CNR, 1996), which have been published on the
occasion of ICME 7 (Québec, 1992) and ICME 8 (Seville, 1996).
The volume consists of two parts.
The first section includes a paper by F. Arzarello on the present trends of
research in Mathematics Education in Italy and a paper by N.A. Malara, dealing
with the history of the “Seminario Nazionale”. The first section also includes
two contributions, by C. Marchini and N. Malara, in memory of Francesco
Speranza, who can be considered an historical father of Didactics of
Mathematics.
The second part is a survey of the Italian reseach in Mathematics Education
since 1994. Abstracts of selected papers, published until june 2000, are given.
We hope this volume can offer an overview of the problematiques which are
mainly addressed by the Italian researchers, in the view of enriching the
international debate and fostering cooperation.
Modena, june 30th, 2000
part one
Part two
RESEARCH FOR INNOVATION IN I TALY: A THEORETICAL FRAMEWORK
Ferdinando ARZARELLO
Introduction
This paper will focus on the most relevant features of Italian Research into
Mathematics Education (RME from now on) since 1960s and of its evolution;
similarities and differences with RME carried out in foreign countries will be
identified. The "top-down" analysis will start with general considerations
leading to a theoretical framework; then significant examples will be
illustrated. Methodological issues that appear particularly intriguing for
current Italian Research will be discussed.
The paper is divided in four parts. Sections 1 and 2 will discuss the
theoretical framework; section 3 will present the current themes of the Italian
research to be analysed within such framework, including some paradigmatic
examples; section 4 will discuss methodological issues.
1. Theoretical framework.
The theoretical framework draws on the analysis of the more recent Italian
research into Mathematics Education which has been carried out by the
author and M. Bartolini Bussi and previously published: see Arzarello (1992,
1996), Arzarello & Bartolini Bussi (1998). More details can be found in
Bazzini & Steiner (1989, 1994), Barra & al. (1992), Malara & Rico (1994)
and Malara (1998).
Such analysis identified four main components of the Italian RME, which are
indicated as A, B, C and D. A and B are the oldest ones: they are
distinguished only for conceptual reasons, but they are usually both present in
the works of a same author (in fact, they belong to the Italian tradition and
they are not explicit paradigms), particularly in the period 1960s to 1980s. In
the last decade, these components have been more integrated than before and
they have interacted with a third component, which is not Italian (component
C); the current research represents an original integration and elaboration of
these three components, so that we can talk about a new trend (component D).
Actually, the first three components may be considered (also) as local
representation of general trends within the international scientific research
community. For example, E. Bishop distinguishes three types of traditions:
the scholastic philosopher tradition, corresponding to component A, the
pedagogue tradition, corresponding to component B, and the empirical
scientist tradition, corresponding to component C.
In the following, the first three components A, B and C will be briefly
discussed.
COMPONENT A
RME based on the conceptual organisation of the subject
(mathematics)
The aim of this kind of research is to improve the teaching of mathematics in
"generic" situations, working on the logical organisation of concepts within
mathematics. Attention is paid only to the contents, while the problems due to
the didactical transposition (cf. Chevallard, 1980), i.e. the relationship
between scientific knowledge and the knowledge to be taught in the
classroom, are not explicitly considered. Didactical interventions in the
classroom are planned taking into consideration students' difficulties related
to mathematics only, and psychological difficulties are usually related to
these ones too. Teaching products, rather than teaching and learning
processes, are the object of study.
A good example of such research is the Syllabus, which was published at the
end of the 70s by the Unione Matematica Italiana (UMI, 1980); it provided a
hierarchic description of the main concepts and abilities that students who
intended to undertake a scientific degree should master at the end of
secondary school, it showed the main difficulties and mistakes and it
proposed challenging problems.
Many mathematicians' contributions to Mathematics Education in the 1960s
reflected this perspective and can be classified in this component: they
advised a didactics based on the conceptual organisation of the discipline.
Examples from the international community are the research carried out by
Servais, Steiner, Rosenbloom (presented at the first ICME Conference held in
Lion in 1968) and some works of the TME group (Theory of Mathematics
Education), especially the old ones.
In the same period however, teachers and other related associations arose new
movements for innovation in Mathematics Education in Italian schools. They
wanted to have access to paradigmatic examples aimed at improving the
teaching of mathematics in specific contexts, in which concrete difficulties
were present. A need for relating the problems of Mathematics Education to
the entire social and pedagogical environment was manifested; a language
more complex and broader than the strictly disciplinary one was requested
for.
Accordingly, a second component was born.
10
COMPONENT B
RME for concrete innovation in the classroom
The aim of this kind of research is the production of paradigmatic examples
for improving the teaching of mathematics in 'specific' contexts, with respect
to concrete and peculiar problems emerging in the everyday life of a
classroom. Intervention in the classroom is rooted in practice and great
attention is paid to teaching and learning processes (not only to products).
The following extract from a document written by Emma Castelnuovo is a
significant description of this component:
11
It is necessary to refer to objects and actions, if the aim is to make the teaching
of intuitive geometry constructive, and as a consequence formative. These
objects and actions must not be predefined, but must change according to the
needs teachers identify in each classroom at different times. The practical means
used in the various experiences have no importance: they can be models, tools,
imagined or implemented experiences about sunlight or shadows. The freedom
to create and interpret, both for teachers and students, is one of the
characteristics of the constructive method1. (Castelnuovo, 1965)
Similar research was carried out in the same period at international level, see
for example Gattegno and Dienes, and most of the works presented at the
CIEAEM conferences in that period.
Even though the movements for innovation were rooted in schools, many
professional mathematicians had a role in this component too, especially
when the Nuclei di Ricerca Didattica2 (NdRD), which had a very important
role in the following years, were created (cf. Malara, 1998): one of the best
known mathematicians who had a fundamental role in this sense is Giovanni
Prodi (cf. Prodi, 1975/77, 1992).
Consequently, it is difficult to classify different authors’ contributions under
one component only: sometimes both components A and B usually belong to
and are integrated in the works of one author. Moreover, in both cases the
research outcomes (i.e. conceptual frameworks in A and experimental
innovative frameworks in B) were not to be directly implemented in the
classroom, but they needed further elaboration: these products were defined
energisers of practice in a ICMI document about the nature and results of
Education research (ICMI document, 1995).
Even though, the two components had different impact on the educational
system. Component A showed a “top-down” model: conceptual analysis
informed and determined practice; component B was based on actionresearch: the starting point was a concrete problem perceived by teachers and
practice was determined by concrete conditions for action in the classroom. In
the first case, the impact on the educational system related to an “intended
curriculum” (curriculum and textbooks, cf. Barra et al., 1992), while in the
1
2
12
Here is the original italian text: "È necessario ricorrere all'oggetto e all'azione se si
vuole che l'insegnamento della geometria intuitiva abbia un carattere costruttivo e che
sia quindi formativo: ecco la conclusione a cui vorremmo aver condotto il lettore.
Oggetto e azione che non devono seguire uno schema prestabilito, ma lasciarsi ispirare
ogni volta dalle esigenze della classe che l'insegnante avrà la sensibilità di saper
cogliere: è proprio da queste esigenze che sono sorti gli esempi che abbiamo dato. I
mezzi pratici per la realizzazione delle esperienze non hanno nessuna importanza: si
tratterà di un modello, di un dispositivo, di un'esperienza realizzata con l'aiuto di un
materiale o solamente immaginata, delle variazioni di una luce o del mutarsi di
un'ombra. Ed è proprio forse questa libertà di ideare e di interpretare, ugualmente alla
portata del maestro e dell'allievo, che costituisce una delle caratteristiche del metodo
costruttivo." (Castelnuovo, 1965, pag.65)
Groups for Research in Education
second case, the impact related to an “effective curriculum” (referring to real
classroom situations). In both cases dissemination relied on an optimistic
faith in teachers: in A, teacher training and in service courses were aimed for
13
(assuming that this was enough to give way to change), while in B
dissemination totally relied on teachers’ capacity and intention to spread the
professional competence acquired through experience, among their
colleagues.
A particular remark concerns the fact that the existence of mixed research
groups (in which both teachers from schools and researchers from
Universities were working together) allowed an increasing integration of the
two components, so that a separation between theoretical research and actionresearch, which took place in other European countries (e.g. France and
Germany, see: Rouchier, 1994, Griesel & Steiner, 1992), was avoided in
Italy.
The integration of the two aspects was fostered by the policy for the
innovation of the Italian curriculum in the 1970s and 1980s, which involved
both mathematicians from University and teachers from primary and
secondary school, who were already involved in the NdRD's (and inservice
courses).
The most interesting examples of such an integration are some secondary
schools textbooks written by Lombardo Radice, Prodi, Speranza, Villani and
others, and the RICME project for primary schools, co-ordinated by Pellerey
(Pellerey, 1979/82).
The collaboration between practising mathematicians and teachers is a
peculiar factor of Italian research and proves essential to understand its
evolution over years (more details and a comparison with other countries in
which some kind of collaboration between Universities and schools is
somehow present can be found in Boero, 1994; further information about the
Italian situation are in Malara, 1997, 1998). At the time of innovation in the
school curricula (at all levels), the NdRD realised they needed to better
characterise the interventions which were implemented in the classroom. In
fact, they realised that neither the analysis based on the conceptual
organisation of the discipline (component A) nor the action-research in a
concrete classroom context (component B) could account for a scientific
explanation of why the 'same' innovation succeeded in some classrooms but
completely failed in other classrooms (a similar discussion with respect to a
non Italian context can be found in the analysis of the IOWO project, given
by Douady, 1988).
This issue proves relevant as far as the research methodology is concerned,
particularly because of the separation between the community of educators
and the Mathematics Education (and science education in general) research
community, which has always existed in Italy as part of a persistent cultural
tradition.
14
However, some important events took place in the 1980s and presented
Italian researchers with new research tools and methods with respect to
Mathematics Education. In particular, we mention:
- the CIEAEM 33, held in Pallanza in 1981 (Pellerey, 1981);
- the four Mathematics Education schools held in Trento in 1980, 1983,
1984, 1991;
- the summer school in Mathematics Education, held in Turin in August
1990.
Thanks to these events, many Italian researchers could interact with wellknown researchers of the international Mathematics Education research
community and came in contact with different methodologies and traditions,
in particular with the methods of what can be defined as component C.
COMPONENT C
RME as observation and modelisation of 'laboratory' processes
The aim is to get a better understanding of the processes that are taking place in the
classroom (in particular short-term processes), in order to plan classroom interventions.
This research requires the use of methodologies that are borrowed from other disciplines,
as psychology, sociology and pedagogy: typically, experiments are prepared, either in a
laboratory or in the classroom which works as a laboratory, in order to test previously
formulated hypotheses. Many research papers published in the PME Proceedings come
from this kind of research.
Typical results from this type of research are for example taxonomies, models
of interactions, etc…, i.e. the products which are defined economisers and
demolishers of illusion in the ICMI study previously mentioned (Kilpatrick &
Sierpinska, 1995).
Component C differs from A and B, in that the starting points for C are
'internal' research problems and its impact on the educational system is not a
main goal: attention is paid to the observation of processes taking place in the
classroom.
The influence of such a component, which had been completely absent in the
previous years, was very important because it was the starting point for a
reformulation of the whole Italian Mathematics Education research and
determined the characteristics of the current RME.
The interaction among the three components (two native ones and one
imported from abroad) produced a very peculiar and original development
and evolution of the Italian RME, which makes it interesting at international
level.
15
2 The Current Trends in the Italian Research into Mathematics
Education
The new developments of the Italian research were discussed at the 8th
National Seminar for Mathematics Education, held in Pisa in 1991 (Cf.
Arzarello, 1992; Boero, 1992; Malara, 1992). Researchers realised that the
component B (innovation in the classroom) was more and more integrated in
the other two components A and C. Consequently, a new trend emerged:
innovation was no longer considered to be only action in the classroom, but to
be research itself. This is the core of the current Italian research into
Mathematics Education.
TREND D
Research for innovation
The object of study is the teaching and learning of mathematics, both in
specific classroom contexts and in relation to the more general educational
context. The research aims are:
(i) to develop paradigmatic examples concerning improvement in the
teaching of mathematics (e.g. partial or global curricula innovations);
(ii) to study the conditions for their implementation in the classroom and the
factors which may be an obstacle to this;
(iii) to develop innovative theoretical models, which may be used by teachers
to guide their action in the classroom;
(iv) to develop innovative methodologies of working in the classroom.
The teaching and learning process is at the same time object and objective of
research.
The first point comes from component A, the second is linked to the
investigation of classroom processes (components B and C), while the last
two points are typical of the new trend (D) and integrate and elaborate
elements from both A, B and C.
Interventions in the classroom are planned with attention to: cultural and
epistemological perspectives (A); pedagogic (B) and conceptual (A) reasons;
cognitive difficulties (C) and specific interactions in the classroom.
The impact on the educational system concerns:
(a) dissemination of the projects for curricular innovation to big groups of
teachers (following the tradition of the NdRD);
(b) discussion of the complex classroom processes, which are object of
investigation, with teachers in inservice courses, in order to make them
aware of their important role in the classroom;
(c) dissemination of methodological innovations, which are suggested by the
research methodology itself.
16
There are a number of different research problems to be investigated. For
example, one main research problem concerns the production of scientific
knowledge about the relationships between projects for innovation (based on
epistemological, cultural and cognitive hypotheses) and their implementation
in the classroom (analysed in different ways). Research for innovation is
characterised by experimental features (inherited by components B and C) to
be integrated with theoretical aspects (typical of A and C). The main aim is
innovation as research and not only innovation as action in the classroom.
Teachers are actively involved in all the phases of research: the method of
participant observation (Eisenhart, 1988) is the most commonly used to
collect and interpret data.
Typical outcomes from this kind of research are: projects for global or local
(e.g. with respect to contents or teaching methods) innovations in the
curricula; models for classroom processes (e.g. the role of the teachers);
etc…. These products are usually framed by a theoretical framework, which
is a result of the research itself (this makes the difference to component B).
Consequently, not only concrete products (linked to concrete contexts) are
developed, but also basic results (the variables which characterise the studied
contexts are made explicit in the research).
This aspect is a result of the collaboration teachers-researchers, both in the
planning and in the observation phase. Such a collaboration has allowed the
distinction between “theoretical” and “practical” relevance of research
(Sierpinska, 1993), which is present in many countries, to be overcome since
the beginning. Theory and practice are generated and develop at the same
time. This methodology of research, which has empirical basis, nowadays
represents a crucial epistemological perspective. Avoiding the distinction
observer-observed (in Education research the observer is represented by the
researcher and the observed by the classroom together with the teacher)
represents a shift from the traditional positivistic methods, borrowed from
natural sciences.
Briefly speaking, the most innovative aspects of current RME in Italy concern
the elaboration of useful tools for dealing with its double-sided products: on
the one side, specific concrete results; on the other side, general and abstract
theoretical results. The internal tension is due to the simultaneous presence of
the three components A, B and C discussed above. In order to give an
authentic account of trend D, a dynamic description of the reciprocal interrelationships established among the three previous components is needed.
Many Italian (and international) researchers agree to consider RME mainly as
the study of mathematics teaching and learning processes as complex
dynamic systems (cf. the notion of complementarity in Steiner (1985) and the
conclusive discussion in Arzarello & Bartolini Bussi (1998)).
17
In conclusion, we see that the evolution of the Italian RME started from
primitive germs and developed complex problems and methodologies: from a
number of first order variables describing the basic components (A, B and C),
a net of new variables connecting the basic ones at a higher level (second
order variables) has been identified in trend D.
Therefore, in order to give account for this two-level complexity which
makes Italian research so peculiar, appropriate tools need to be created. Two
tests for the analysis of Mathematics Education research were developed.
The first test is
TEST 1 (MINIMALITY)
Educational research must contain at least two of the three components A, B
and C. In order for it to be relevant, all of them must be present.
Let us consider some examples.
a) Boero & Szendrei (1995) show the necessity of considering not only
quantitative aspects but also “qualitative information regarding the
consequences of methodological or content innovations”. This is a way of
considering second order properties. Other second order variables are
mentioned by the authors: “the relationships among teachers, students and
mathematical knowledge in the classroom; between school mathematics
and the mathematics mathematicians do; between research outcomes and
classroom practice”.
b) The reconstruction of the development of education research in France
presented by Perrin-Glorian (1994) shows the use of concepts which are
elaborated within different cultural and conceptual frameworks and
drawing on different problems; consequently there is the need for a second
order analysis in order to elaborate an appropriate comprehensive
theoretical framework.
It is important to say that a second order analysis does not simply combine
the first order components like in a jigsaw, but these variables are related to
one another within a system. The idea comes from Vygotsky (1990): he
advocates for the necessity of studying single components of a phenomenon,
which still have the same characteristics of the global phenomena, without
reducing the phenomena to too fine components which have lost the global
features. (Vygotsky presents the example of a molecule of water and the
atoms of hydrogen and oxygen). In the same way, in RME it is worth
studying conceptual hierarchies for mathematics and for social interaction
within a context in which they interact.
18
Given that a research project passes the test for minimality, a second test
needs to be undertaken.
Such approach to RME is an elaboration of the notion of complementarity
presented by Steiner (1985): cf. the discussion in Arzarello & Bartolini Bussi
(1998).
TEST 2 (DYNAMIC INTER-FUNCTIONALITY)
Dynamic inter-functionality among its components must be satisfied in
Educational research. That is, the analysis must concern the relationships
between components more than the components themselves.
To summarise, many Italian researchers agree to say that doing RME
nowadays means investigating teaching and learning processes in
mathematics considered as complex dynamic systems. That is, they are (or
should be) analysed as processes in which all the components live together in
the concrete context which is studied. This new approach is currently being
developed in Italy and it has got very important consequences both at the
level of innovation in content and at the level of methodological innovation.
Current Italian RME consists of an effective interaction between projects for
innovation (based on epistemological, cultural and cognitive solid
hypotheses) and their implementation in the classroom (to be analysed in
various ways). A key point of such an interaction is the active involvement of
teachers in all phases of research as participant observers, as previously
mentioned.
The following sections will discuss this perspective, presenting concrete
examples and analysing the methodological consequences.
3. The Current Themes in the Italian Research into Mathematics
Education
The current themes in the Italian research into Mathematics Education can be
classified in two categories.
(i) Contents:
1. Geometry
2. Algebra
(i) Cross-curricular themes:
19
1. Social Construction of Knowledge
2. Real Contexts
3. Theory and Theorems
4. History and Epistemology
5. Learning Difficulties
6. Beliefs
7. New Technologies and Multimedia
The themes in the second category are the core of the current Italian research.
It can be argued that a good understanding of these themes is possible only if
both first order and second order variables are taken into consideration.
Most of the cross-curricular themes can be seen as:
1. Derived from the components A, B and C, within a first order analysis.
2. Interaction of the components A, B and C to produce more complex
themes, within a second order analysis.
This situation means a peculiar analysis of the methodology has to be carried
out, with respect to the two levels (see section 4).
SECOND ORDER VARIABLES
T
H
E
M
E
S
1
SOCIAL
CONSTRUCTI
ON OF
KNOWLEDGE
2
REAL CONTEXTS
3
THEORY AND
THEOREMS
4
HISTORY AND
EPISTEMOLOGY
.
Mathematical
Discussion
Semiotic
Mediation
(gestures)
Discussion
Argumentation
Proof
Reading
Interpreting
Historical
documents
(voice-echo)
2
Fields of
Experience
Cognitive
Unity
3
Dragging
Theorems
1
Table 2
Analysis/
Synthesis
Both theme 1 (Social Construction of Knowledge) and theme 5 (Learning
Difficulties) are strongly connected to component B (innovation in the
classroom).
20
Theme 2 (Real Contexts) refers to component B as well (see for example, the
works by E. Castelnuovo and the project about the mathematisation of reality,
cf. Spotorno & Villani
Theme 3 (Theory and Theorems) refers to component A (a concept based
didactics).
Theme 4 (History and Epistemology) derives from A as well (see the works
by Enriques and L.Lombardo Radice).
Theme 6 (Beliefs) comes from a 'non Italian' tradition, within component C
(observation of laboratory processes), but it has got a strong impact on the
Italian tradition (e.g. the teacher-researcher).
A separate discussion concerns theme 7 (New Technologies and Multimedia):
this cannot be directly matched with one specific component; it appears to be
a ramification of theme 2 (this hypothesis needs more justification), i.e. the
problem of new technologies is considered as part of the real context of
modern society (more details can be found in Malara, 1998)
However, it must be remembered that the current research projects centred
around these themes must be analysed according to second order variables, as
the interactions among all the different components need to be considered in
order to understand such a complex situation.The (a-priori and a-posteriori)
analysis of mathematical knowledge, of didactic variables, of students and
teachers' mental dynamics and teachers behaviour in the classroom can be
expressed only at a second order level, dynamically and inter-functionally.
In order to make this clear, some paradigmatic examples are presented. The
table shows some intersections of different themes, which are at the basis of
the second order developments of research. The rows and columns contain
some of the cross-curricular themes. The diagonal contains the core issues to
give rise to second order trends, with a few intersections represented by the
cells outside the diagonal. For example, the intersection of theme 2 (Real
Contexts) and theme 1 (Social Construction of Knowledge) produces the
Vygoskian research, which investigates the complex interaction of tools,
gestures and language in the construction of knowledge. Examples are the
voice-echo game, presented by Boero et al (1997, 1998) and the project about
Gears in primary schools, carried out by Bartolini Bussi et al (1999).
In the following, some other examples will be briefly discussed.
The theme Theory and Theorems (column 3) concerns the study of
epistemological aspects of proof within mathematical contexts (geometry,
arithmetic, etc…), as well as the analysis of cognitive aspects which make
proof become object of didactics. The cognitive and didactical analysis makes
new concepts emerge. For example the concept of Cognitive Unity (see
Garuti et al., 1996, 1997, 1998) and the voice-echo game are second order
21
concepts, both because they involve an interaction of different components
(mathematical concepts and historical analysis, cognitive aspects, social
construction of knowledge) and because they require an integration of
complementary methods.
Based on the historical Italian tradition, which is characterised by studies
about the foundations of mathematics and their impact on education (see
Enriques, with respect to component A and Castelnuovo, with respect to
component B), more recent studies on the same subject (Arzarello et al.
1998a, 1998b; Bartolini Bussi et al., 1996, 1999; Mariotti & Bartolini Bussi,
1998; Boero et al., 1998) illustrate an original approach to research on the
teaching and learning of theorems, in which a second order analysis is crucial
(trend D). In fact, the characterising features of current research are the
identification of the features of theorems that have not changed over time
(from Euclid till now) and the experimental investigation of students'
appropriation of theorems, in a holistic perspective, which does not separate
the phase of constructing a proof from the other activities related to theorems:
the production of conjectures, of definitions, etc…
These issues are intersected (trend D) with another theme which is part of a
very old historical and epistemological tradition (component A), which goes
back to the XIX century, and is still linked to an extensive presence of
geometry in nowadays curricula at all Italian school levels (at least in the
scientific schools).
An interesting example of how trend D has emerged from the other
components A and B is the ICMI Study Conference held in Catania,
presented in a volume edited by Mammana & Villani (1998).
The characteristics of Italian research about the teaching and learning of
geometry are: historical and epistemological analysis of curricula,
experimental investigations in schools, attention to epistemological and
cognitive key features of geometrical reasoning (related to first order
components). The innovative analyses and projects elaborate these issues
identifying second order components. See for example: Malara (1997, 1998)
with respect to the difficulties in the learning of geometry; Bartolini Bussi et
al. (1994, 1998c, 1999) with respect to the construction of mathematical
knowledge; Mariotti et al. (1989, 1997, 1998) with respect to figural and
conceptual issues and the role of theory in the construction of theorems.
Similar discussion concerns the theme of the contextualised teaching of
mathematics (cf. column 2, Real Contexts in Table 2), which is typically
cross-curricular and it is linked to the problem of the development of
mathematical reasoning in relation to reality, that is a central problem in
many Italian works. This theme goes back to the massive didactical
innovations of the 70s (component B, cf. Castelnuovo). Starting from first
22
order research, new original second order contributions, concerning
classroom work, have been produced. Boero elaborated the concept of field of
experience, which is connected to other themes (Table 2, line and column 2).
See also the works by Bartolini Bussi (1996; 1999), Boero et al. (1995),
Basso et al. (1998), Dapueto & Parenti (1999), Lanciano (1996).
Another second order theme is the Social Construction of Mathematical
Knowledge in the classroom (Table 2, line 1). This theme derives from the
classroom experiments developed in the 1970s (component B), which
provided a relevant contribution as far as the involvement of teachers in the
research projects carried out by schools and universities together. As a
consequence, the image of the teacher-researcher emerged (cf. Navarra & De
Plano, 1992). An evolution towards second order research has taken place in
particular regarding the investigation of the approach to theoretical thinking
and to mathematical reasoning: see Bartolini Bussi (1996, 1998a, 1998b) and
Pesci (1998).
Another example concerns research about teachers and students beliefs, and
their mutual interactions. This theme is currently being developed in Italy
with specific features (in some way related to international research; see:
Cannizzaro, 1989; Furinghetti et al., 1990) and it still needs further
investigation. The theme proves relevant for a better understanding of the
teaching and learning reality in schools, as the basis for planning innovation
in the classroom. Italian issues, as for example the problem of the teacherresearcher, are linked with non-Italian issues (related to component C). The
following excerpt from Malara (1999) illustrates the first problem:
Sometimes teacher-researchers constrain the choice of topics, which can be
object of innovation in the classroom…. As far as methodology is concerned,
teachers’ autonomy often does not allow having common frameworks for the
classroom discussions, recording all the discussion, having a detached
observer in the classroom…. Research results strictly depend on the kind of
relationship existing between the director of research and the teacherresearcher3.
Recent Italian research around this theme (see for example Bottino &
Furinghetti, 1991; Poli & Zan, 1999) has shown typical second order features,
e.g. the study of teachers conceptions to be related to students conceptions
about some key points of the mathematical curriculum and its recent
3
“[L’insegnante-ricercatore spesso pone] seri vincoli nella scelta degli argomenti di innovazione
su cui innestare le attività sperimentali…Sul versante metodologico, spesso l’autonomia li porta
a dare poca attenzione alla pianificazione comune dei canovacci di discussione; a limitare la
registrazione delle discussioni…; a rifiutare l’intervento di un osservatore muto…i risultati di
ricerca sono strettamente dipendenti dalla sintonia e dal delicato equilibrio con cui si gioca il
rapporto tra insegnante ricercatore e direttore della ricerca" (Malara, 1999).
23
transformations (e.g. the integration of new technologies in the teaching and
learning of mathematics and in the work of mathematicians).
Further examples concern the themes in (i), in particular the works about the
teaching and learning of algebra and the analysis of its links with arithmetic.
Interesting results have been produced in Italy and appreciated at
international level. See Arzarello (1998); Arzarello et al. (1993, 1994a,
1994b, 1995; 1999, 2000); Bazzini (1999); Boero (1999, in print); Malara
(1999, in print); Menghini (1994); Reggiani (1997, 1999); Basso et al. (1998)
and the included references. This theme has roots in the works by F.Speranza
(component A), E. Castelnuovo (component B) and G. Prodi. The transition
from the components A and B to second order analysis is marked by a
number of events: the participation of some Italian researchers in the Algebra
Discussion Group at PMEs; the WALT Seminar (Workshop on Algebraic
Learning in Turin, held in 1992 in Turin), which well known international
researchers participated in (A.Sfard, A.Bell, R.Sutherland) and which SFIDA
(Séminaire Franco Italien de Didactique de l’Algèbre) originated from, in
collaboration with J.P.Drouhard (cf. Drouhard & Maurel, 1995). The analysis
of the complex dynamics which originate from creating, manipulating and
interpreting formulas gave rise to the development of theoretical models
(Arzarello et al., Boero), interventions in the classroom based on those
models (Malara, the research group in Pavia), sometimes also introducing
new technologies (Chiappini, Lemut) and historical and epistemological
investigations about algebraic language (Menghini). The common feature to
this research is that the elements coming from components A and B are
integrated in a more complex theory at a second order level: a paradigmatic
example is the notion of ‘space-time of production and action’ described in
Arzarello et al. (1995).
A final remark. Multidimensional analysis is nowadays very often required.
For example, in the experience of gears and circles developed by Bartolini
Bussi, the reference to Erone requires a link with the theme History and
Epistemology; the research about dragging, carried out by Arzarello, Olivero
& Robutti (cf. Arzarello et al., 1998a; 1998b) is related to the theme of
Semiotic Mediation, gestures, Theory (as dragging is studied in the context of
theorems production) and History (as the problem is connected to the notions
of Analysis and Synthesis in geometry).
4. Methodological Issues
The Italian framework emerging from the previous description appears
fragmentary. This is not only due to the fact that a number of works from
24
different people have been presented but also to the fact that the problems
investigated are different in structure. E.g. the problem of modelling is
different from the issue of theorems, and so on.
Such variety (and richness) of problems has deep consequences at
methodological level. In fact, a contradictory and heterogeneous
methodological framework results from that. This can have two meanings: (a)
the development of research has yet not achieved a good elaboration as far as
methodology is concerned; (b) the nature of the problems investigated is
contradictory itself, so that their study may require a number of different but
complementary methods, which are not always coherent (cf. Steiner, 1985).
This section will focus on some issues concerning this problem: these are
preliminary questions to be further formulated and researched.
a) Student’s time, teacher’s time and researcher’s time.
A new variable is to be taken into consideration in order to make sense of the
second order phenomena previously described: the time variable. Second
order variables are significant because they give an account of the real
temporal development of events in the classroom. This temporal development
takes place with different velocities and some particular phenomena may be
fully understood only if what happens in the micro-time is connected to what
happens in the macro-time (cf. Boero, 1990). Second order variables often
involve the use of first order variables, which have different ‘natural times’:
this is one of the causes of the methodological problems. To say it in a
metaphor, the situation is similar to a physical phenomenon which needs to be
dealt with from the point of view of both classical mechanics and quantitistic
mechanics, which may be in contradiction. There is a problem in studying
second order didactical phenomena: they need to be studied according to both
a fine and a global analysis, so that they seem to require the use of a number
of different research methods, always complementary, sometimes
contradictory. On the one hand, there are long-term processes, typically
connected to innovative problematiques within trend D (originated from B),
based on the analysis of the evolution of students’ processes, of teachers’
beliefs, etc…in macro-situations. On the other hand, there are short-term
processes, linked to trend D as well (with origins in component C), based on
the fine analysis of micro-situations.
Long term processes are not simply the sum of short term processes: the
change in the time according to which these two kind of processes take place
implies a change in the methods used to observe what happens in the
classroom.
25
b) Status of mathematical objects
The peculiarity of Education research provides motivations for reconsidering
(redefining) the epistemological status of the mathematical objects taking part
in the teaching and learning processes. The mathematical objects must be
considered in relation to the cognitive, social, etc…processes under which
students construct (or are not able to construct) these objects, within the socalled teaching & learning problem situations. There are a number of possible
answers to this problem; they are very different and often complementary,
and they are defined by explicit or implicit didactical theories (which
characterise the teaching and learning situations). Moreover they are deeply
related to the temporal issue mentioned in a): the different observations used
according to the change in the time imply a change in the status of
mathematical objects and concepts (such a consideration completely change
the first order framework, e.g. related to component A).
c) Scientific aspects of the knowledge related to didactical innovation.
The consequence of some peculiar aspects of the Italian RME – typically, the
presence of teachers as participant observers, the refusal to do 'laboratory'
research, the didactical continuity provided by the same mathematics teacher
teaching each classroom over long periods of time at all school levels – is that
real teaching contexts are investigated, and learning objectives are usually
considered more important than research aims. Therefore, the temporal
dimension of all the variables used has to come to terms with the real time of
the concrete classroom observed. This is a heavy contradiction, that is
however essential in real teaching contexts: in a laboratory the time variable
can be contracted or dilated as wanted, whereas in a real classroom this is
more difficult. This approach puts under trial the ‘laboratory’ methods typical
of component C and makes it necessary to redefine the problem of the
'scientificity' of the Italian research (cf. Boero & Szendrei, 1995). The
problem of reproducibility and quantification of results needs to be tackled
from a different point of view. A hypothesis is the following. While
quantitative methods and reproducibility criteria are fine for the first order
variables, as it is possible to consider, measure, etc…them according to a
single framework, they no longer work for second order variables, as multiple
frameworks are needed. A sort of indetermination principle for didactic
variables is established. The classical scientific methods can no longer be
useful in this new framework.
Within such a complex and not yet well-defined framework, researchers
normally use a number of different methods, which appear to be functional to
the problem investigated.
26
For example, as far as the study of the role of the teacher is concerned,
Activity Theory seems to be the best approach; on the one hand, because the
Vygotskian tradition has given consideration to teachers, on the other hand,
because Leont’ev framework seems to be particularly helpful in analysing
and planning the role of the teacher (Leont’ev, 1978). This line of work is
followed by M.Bartolini Bussi (cf. the project about gears and circles),
M.Alessandra Mariotti (the Cabri project), F.Arzarello (primary school). A
remark: provided the didactical contract is fulfilled, in some cases a dilation
in the normal time of the classroom happens, due to the need of studying
carefully the different dynamics taking place during the classroom
discussions (this has been observed both in Modena and in Turin).
The role of the teacher may be studied with respect to other second order variables, e.g.
his/her beliefs system (cf. Furinghetti, 1998).
Changing variables in the research may prove the methodology of Activity
Theory no longer useful. For example, a microanalysis of students’ times
when doing individual problem solving or when interacting with the teacher
(cf. Boero & Scali, 1996) may require elements gathered from different
theoretical frameworks, but complementary to one another (this shows the
non solvability of the indetermination principle, at the moment). For
example, the group working in Genova (Boero) refers partly to the theory of
didactical situations (with some changes in respect to the interplay between
the action and formulation phases), partly to the Vygoskyan notion of
mediation, which is extended to the whole process (and not only to the phase
of institutionalisation) and partly to its own theory (cf. the voice-echo game).
A remark: in this case there may be a contraction of the real time in order to
have more compact observations of the students’ processes to undergo the
micro-analysis. Such phenomenon is the opposite of the dilation in time
mentioned before with respect to the Activity Theory framework
In a similar way, other problems may underline the necessity of considering
second order variables to be analysed in their interplay with the status of
mathematical object over history. A typical example is the study of abductive
phenomena in the context of theorems production (Arzarello et al., 1998a);
from a phenomenon to be studied from the cognitive micro-time of students,
it becomes an epistemological problem to be studied in the historical context
(Arzarello, 1999), with reference to the ‘analysis and synthesis’ methods in
geometry.
Another similar example is the investigation around gears carried out by
Bartolini Bussi (Bartolini Bussi et al, 1999).
Acknowledgements.
I would like to acknowledge M. Bartolini Bussi, P. Boero and N.A. Malara
for their criticism, suggestions and comments during the writing of this paper.
27
A special thanks to Federica Olivero, P.L. Ferrari and N.A. Malara, without
whose help and encouragement this paper would not exist.
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Didactique des Mathmatiques en France: Hommage à Guy Brousseau et
Gérard Vergnaud, 97-147, La Pensée Sauvage, Grenoble.
PESCI, A.: 1998, 'Class discussion as an opportunity for proportional reasoning',
Proc. PME 22, Stellenbosch, Vol. 3, pp. 343-350.
POLI, P., ZAN, R.: 1999, 'Winning beliefs and mathematical problem solving',
Proc. CERME-I, Osnabrueck (in press).
PRODI, G: 1975/1977, Matematica come scoperta voll. 1 e2, D'Anna, Firenze.
PRODI, G.: 1992, Ricerca in Didattica della Matematica, Notiziario dell'Unione
Matematica Italiana 19 (1-2), 146-150.
REGGIANI, M.: 1997, Continuità nella costruzione del pensiero algebrico,
Notiziario UMI, suppl. n. 7, 35-62.
REGGIANI, M.: 1999, 'Syntactical and semantic aspects in solving equations: a
study with 14 year old pupils', Proc. CERME 1, Osnabrueck (in press)
ROUCHIER, A.: 1994, Naissance et Développement de la Didactique des
Mathématiques, in Artigue, M. et al. (eds.), Vingt ans de Didactique des
Mathematiques en France: Hommage à Guy Brousseau et Gérard Vergnaud,
148-160, La Pensée Sauvage, Grenoble.
SIERPINSKA, A.: 1993, Criteria for Scientific Quality and Relevance in the
Didactics of Mathematics, in G. NISSEN, M. BLOMHOJ (eds.), Criteria for
Scientific Quality and Relevance in the Didactics of Mathematics, 35-74,
Danish Research Council for the Humanities: The Initiative 'Mathematics
Teaching and Democracy, Roskilde.
SPOTORNO, B., VILLANI, V.: 1972, Mondo reale e modelli matematici, La Nuova
Italia, Firenze.
STEINER, H.G.: 1985, Theory of Mathematics Education: an Introduction, For the
Learning of Mathematics 5 (2), 11-17.
TREFFERS, A.: 1987, Three Dimensions. A Model of Goal and Theory Description
in Mathematics Instruction. The Wiskobas Project, Kluwer Academic
Publishers, Dordrecht.
UMI (ed.): 1977/78, Matematica come scoperta: guida al progetto di insegnamento
della matematica nelle scuole secondarie superiori, Esperienze dei nuclei di
ricerca didattica, voll.1 e 2, D'Anna, Firenze
UMI: 1980, Syllabus, Notiziario dell'Unione Matematica italiana 7 (3), 5-16
VYGOTSKIJ, L.S.: 1990, Pensiero e linguaggio, Italian translation of Thought and
Language, by Mecacci, L., Laterza, Bari (first edition in Russian 1934).
ZAN, R.: 1997, Mortalità universitaria e mortalità matematica, Tracciati n. 2, http:
www.eurolink.it/scuola/tracciati.
ZAN, R.: 1998, A Metacognitive intervention in Mathematics at UniversityLevel, in
stampa su Proc. ICMI Studies "On the teaching and learning of mathematics at
University Level”, Singapore (in press).
32
THE "SEMINARIO NAZIONALE":
AN ENVIRONMENT FOR ENHANCING AND REFINING
THE ITALIAN RESEARCH IN MATHEMATICS EDUCATION
Nicolina A. MALARA
Il sole lentamente si sposta
sulla nostra vita, nella paziente
storia dei giorni che un mite
calore accende, di affetti e
di memorie. 4
(A. Bertolucci, At home, 1951)
Italy has an ancient tradition of studies in mathematics education. However,
projects on this field started being sponsored academically only at the
beginning of the Seventies, when the Bourbakist revolution gave vent to
thinking trends in maths education (see Barra et al. 1992).
Ever since the early Sixties, when middle school was unified and became
compulsory, the teaching of mathematics was joined to that of the other
sciences. The aims of this association were: to give more importance to the
cultural dimension of science, which for a long time had been considered
culturally inferior to les belles lettres, and in particular to give a new image of
mathematics, which until then had been disturbed by an extremely technical
teaching. This new combination should show that mathematics has two main
features: it is the language for all sciences, which nourish it, and it is an
independent theoretical corpus (De Finetti 1964, Viola 1965).
In those years, also thanks to scholars such as E. Castelnuovo, the most
widespread vision of mathematics was based on reality. It consisted in
starting from pupils' concrete activity, possibly with ad hoc didactical tools,
and going on through problems, the solution of which could help teachers to
build up and give significance to mathematics at theoretical level.
This was the point where the teacher's role must be transformed: a teacher
could no longer simply carry out the annual syllabuses, step by step, as they
were written; on the contrary, teachers must become designers of long-term
didactical projects, based on precise cultural choices which included a deeper
analysis of specific issues contained in the syllabuses.
4
The sun slowly moves/ over our life, in the patient/ tale of days that a gentle/ warmth
lights up, with love and/ memories.
33
These novelties, introduced by legislation with the new syllabuses for middle
school (1979), made sure that academy promoted many projects so as to give
teachers the awareness of their new role and start a real process of cultural
innovation at social level.
It was chosen to work beside teachers, according to a minor but ancient
tradition of co-operation (Barra et al., op. cit.). The belief was that dialogue
between the parts could give teachers new cultural and methodological hints
as well as awareness of the main obstacles against innovation (teachers'
difficulties and needs, influence of new technologies, cultural values arising,
stiffness of the system, etc.). This was thought to be the best way to find
solutions.
Some universities created the first Nuclei of didactic research (Genoa, Parma,
Pavia, Pisa, Rome, Trieste), and the first Internuclei meetings took place as
seminars. The first studies carried out aimed at contents renewal: they
worked out specific teaching projects with the co-operation of teachers. The
pioneer projects created by P. Boero for middle school, for example, are
global and strongly innovative.5
The books by F. Speranza & A. Rossi dell'Acqua (see here the reports
devoted to Speranza) and the book edited by G. Prodi were born in this
environment. The latter was the result of a specific project between
universities, sponsored by the CNR (National Research Council). Teachers
co-operated to test the effectiveness of the suggestions made in the project.
Then, in the early Eighties, a revision of the syllabuses for the other school
levels was being planned, home computers and programming languages were
spreading, and wide financial investments were made to promote mathematics
education. New Nuclei of didactic research were born, with consequent
widening of the studies realized. Born as small seminars, the yearly
Internuclei turned into national conferences. Many researchers started going
to international conferences,6 and foreign colleagues started being invited to
spend some time in Italy (Malara 1998).
Each Nucleus worked its own way, according to the local situation. Still, the
two main philosophies were: renewal through continuity (attention given to
the usual teaching tools and methods, to the quality and updatedness of
textbooks; innovation suggested on specific issues, mainly crucial ones) on
one hand, and renewal through breakthrough (stop with traditional teaching,
yes to the integration of new technologies and other sciences into the teaching
of mathematics, focus on mathematization and on issues with strong social
relevance, such as probability, statistics and computer science) on the other.
Anyway, both approaches aimed at recovering the historical-epistemological
aspects of mathematics.
5
6
34
See Boero et al (1979), Boero & Guala (1981), Boero (1981, 1985a, 1985b, 1986).
See for instance the proceedings of CIEAEM Pallanza (1981), Orleans (1982) and
Leiden (1985).
These different approaches lead to an inner dialogue in the discipline, in order
to define the general aims, the way they should be tackled (pertinence,
coherence, rigour) and their scientificity, also with reference to the
international panorama.
This was the moment in which the idea of a permanent seminar arose, with
the aim of promoting and refining the Italian research in, creating a common
style of work, finding out research issues to face together and getting out of
cultural isolation and achieving a specific identity within the community of
mathematicians. G. Prodi played here a very important role both from
scientific and political point of view.
In May 1986, a document produced by the members of the nuclei of research
with the support of the CIIM (Italian Commission for the Teaching of MatheExcerpt from the document presented by M. Polo at the first session of the
National Seminar
Maths education as a scientific discipline is being built right now and I think that
first of all we have to analyse the difference between "observation" and
"experiment" as research methods. I am speaking of the deep distinction that
experimental sciences make between the two "momenta". It might seem an
artificial distinction, but indeed it is fundamental because it concerns the study
phase that must take place before any experiment. Only after a priori explicit
hypotheses it is possible to construct and carry out an experiment to highlight
and study the functioning of a given phenomenon; which also implies that these
hypotheses, or part of them, can be discussed also in a second phase, during the
analysis and "study" of the results of the experiment.
Indeed, even the mere observation of a phenomenon happening can be useful to
widen ist knowledge; but "making an experiment" on a given phenomenon
without the right analysis methods doesn't necessarily mean to consequently
"widen" its knowledge.
So, we have to analyse how we have constructed our experimentation till now,
which methods and tools have built the structure of our researches. I think that
this work is necessary, because it is the first step towards understanding
ourselves and being understood, and it makes us more reliable, so that all we say
and write has a common root.
...
Without the will to understand and be understood, we are not going to go any
further. We would stop, because the whole heritage of experience and the
precious work we have done cannot be wasted, if we believe in our profession of
researchers and in the meaningful commitment that we have in front of all the
social components involved in our work.
matics)7 suggested the creation of a "National Seminar for Research in
Mathematics Education". Such document was discussed, modified and
7
CIIM was born in 1908 from within ICMI, and it became part of the Italian
Mathematical Union (UMI) in 1975.
35
approved in a special meeting held in Pisa on June 21, 1986, which can be
considered the birthday of this institution.8
The foundation document illustrated the structure of the seminar: six-monthly
seminar sessions held by a presenter, with the participation of a reactor and of
experts from outside the didactic community. The task of choosing the
presenters and organizing the seminar sessions was up to a committee elected
every year from among the participants to the seminar.
The first committee, which founded the seminar, was constituted by F.
Arzarello, P. Boero and E. Gallo. It organized two session devoted to
researches in the field of compulsory school. The first one was theoreticalexperimental and concerned measure in elementary school9 (presenters: M. G.
Bartolini Bussi and her staff). The second one concerned probability in
middle school10 (presenters: A. Pesci and M. Reggiani), and it was rather
oriented at creating didactical units for a three-year teaching project on this
theme (for more details on this and other seminars, see the summaries further
on).
In order to understand the kind of problems involved, it is very interesting to
read the excerpt of the document that M. Polo, reactor, presented at the first
seminar session with the aim of promoting a collective reflection on research
methodology, value and scientificity.
The second committee,11 constituted by M. Barra, P. Boero and M. A.
Mariotti (1987/88),promoted two other sessions on primary school: one about
the problems of teaching and learning geometry (presenter: M. Polo)12: the
other, theoretically quite complex, about the cognitive and metacognitive
aspects of problem solving (presenter: F. Arzarello).13
Despite their structural differences, the researches presented had something in
common: they both presented theorizations starting from the results of
experimental studies carried out under specific research hypotheses. This was
a novelty in the researches that had been carried out so far (mainly
phenomenological ones), which testifies the evolution in research
methodology, offering a model for the whole community.
The third committee, composed by F. Arzarello, M. G. Bartolini Bussi and M.
A. Mariotti (1989/90), decided to organize two sessions on completely
different themes: the first was devoted to the introduction of the organizing
concepts of computer science into compulsory school (presenter: M.
8
9
10
11
12
13
36
This document was used as reference model by the commission for the manifesto of
the European society of researchers in mathematics education.
The work presented was then published in Bartolini Bussi (1990), Bandieri (1987),
Beretta & Andreini (1987), Tioli (1987).
The work presented is now in the volume Pesci & Reggiani (1987).
Ever since then, it was decided that one of the components was re-elected to create a
sort of continuity
This presentation is reported in Polo (1988, 1989).
A synthesis of this work was presented at the international conference PME 13 in
Paris (1989), see Arzarello 1989.
Fasano)14; the second one, concerning middle school (presenter: P. Boero),
was a critical-theoretical analysis of the results of experimentations belonging
to projects of long-term curricular innovation which had been carried out by
the presenter. There experiments were strongly connected to specific real
contexts.15 For such analysis, focused on the problems of conceptualization
and linguistic mediation, Boero introduced the concepts of "experience field"
and, more in general, "semantic field",16 which today are quite frequently
referred to, but they were totally new then, so that the participants discussed
them for a long time.
This seminar, which dealt with quite complex themes, was a meaningful
contribution for the researchers community because of the many researches
that are at basis of it (see the relative file), involving many fields and opening
to the Vygotskijan thought.
A new committee, constituted by C. Bernardi, P. Boero and E. Gallo, started
its work in 1990, and initially it respected tradition by asking M. A. Mariotti
to present her studies on the interaction between images and concepts in
geometrical reasoning.17 The second seminar, however, was something
different: the community was offered a moment of reflection with a session
on the meaning of didactic research and of its social impact (feedback in
teachers' training, at-large useability of the researches, etc.).
Unlike the past, this time the session was held by many presenters, in that it
offered two round tables: one concerning the relationship between didactic
research and teaching (presenters: N. A. Malara, C. Mammana, V. Villani),
the other discussing the status of Italian research till then (presenters: F.
Arzarello, P. Boero, B. Scimemi). There were also a presentation by M.
Pellerey on the main research models in foreign maths education, a
presentation by the teachers M. Rocco, C. Testa and M. Trevisan on the role
of teachers and pupils in research, and a general presentation by G. Prodi.18
Many debates arose, during this seminar, to discuss how it is possible to
promote a stronger impact in schools and obtain the right acknowledgement
by the institutions of education. The theory-praxis conflict emerged right then
for the first time, and some researchers, like Bartolini Bussi, supported the
14 The essential elements of this presentation are reported in Fasano (1989).
15 The results of this seminar were then discussed by Boero in his plenary conference at
PME 13, Paris (see Boero 1989).
16 Boero writes: "the definitions of 'experience field' and 'semantic field' given in this
report are still approximate: they should be explained most of all by the examples, but
on the other hand they are so 'fluid' because the more examples (and 'episodes' and
'cases' connected) I consider, the concepts of 'experience field' and 'semantic field'
evolve.
17 Some of the results of this seminar can be found in Mariotti (1989) and Mariotti et al.
(1987).
18 Some contributions to the round tables have been published, see Arzarello (1992),
Boero (1992), Malara (1992), Prodi (1992).
37
possibility of making a didactic research not necessarily conditioned by the
question of social feedback..
This seminar, with such wide discussions, was a turning point in the history
of our community: it realized the importance of the work carried out in a
decade, the role of teachers, some of whom had actually become researchers
in the meantime,19 and it could see very clearly that the researches were
evolving after a quite complex methodological model, which in the end is
typical for Italian research. This model considers simultaneously the
mathematical issues in their historical and cultural dimension, the innovative
pedagogical questions of verbalization (written and oral) and classroom
discussion and teachers' training.20
It was then decided to make a yearly seminar and to keep the same committee
for two years.
Also thanks to the results of the latest discussions following committee,
constituted by N. A. Malara, C. Marchini and C. Morini (1992/993), started a
new phase in the planning of the seminar session: the choice was no longer to
have single presenters who would talk about their own researches, but rather
to assign a research theme to a group of researchers. This was meant to
promote the co-operation among researchers coming from different cultures
and areas (which until then had not been pursued) and to strengthen the
researches in the field of secondary school, which until then, owing to
specific historical ad social reasons (see table 2), had been weaker than the
others.
The presenters invited were asked to give a precise overview of the status of
international research, with special attention to new trends and
methodological
Excerpt from the final survey on the management of the National Seminar
in 1992/93
[...] Just a few words to say how we have worked, or better, to show the goals we
have chosen to pursue. […] The first goal was to make the management of the
Seminar open to any suggestion coming from the community, since we intend to
make something for the growth of the community itself. Our tools have been the
questionnaires and the meetings between the members of the committee and the
directors of the various nuclei. Moreover, we have tried inviting as many experts
as possible to the sessions, in order to give more room to dialogue and a
possibility of confrontation among the different voices and styles. We have
offered surveys and reviews, in order to give to the researchers in didactic a
panorama of the present trends on the themes chosen, thanks to the experience of
those who work with these themes.
[…]
One of the ideas that we have not enacted was to manage the seminar through a
call for papers. We of the committee wanted to carry out this innovation, and in
19 Navarra & Deplano (1992) is a document of those times about the nature and role of
teacher-researchers.
20 More on these questions can be found in the paper by Arzarello in this book.
38
the meantime we didn't want to become judges of the products that could be
presented. So prudence guided us...
aspects, so that the participants could be inspired as to new themes for future
researches. The themes presented concerned two traditional teaching areas,
i.e. algebra and geometry.
The first seminar contained the joint contribution of F. Arzarello, L. Bazzini, and G.
Chiappini for middle school, where they presented an interesting teoretical model on
the development of algebraic thought, which is still a reference model for all
Italian researchers dealing with didactic of algebra.21. As to secondary
school, the individual contributions by E. Gallo, F. Furinghetti and M.
Menghini concerned respectively: the problem of literal calculus, the
approach to the concept of function and semantic-syntactical questions
connected to the use of the algebraic language, also with reference to
history.22
The novelties in this seminar were: a) an important joint work of theoretical
research, carried out especially for this occasion; b) the evident necessity that
reactors belong to the research field of maths education, so that they are able
to seize the value of the researches in the international context and make
useful considerations for the presenters.
The seminar about geometry23 focused on epistemological aspects, and it was
held by F. Speranza who, for this occasion, analysed the evolution of the
concept of space according to the points of view of the various ages (see the
summary file or, more widely, Speranza 1994). Other important presentations
came from M. G. Bartolini Bussi, E. Gallo and M. Menghini; the first two
concerned the methodologies used in their researches (carried out respectively
in primary and in secondary school),24 the third regarded epistemological
questions and their influence on the teaching.25 Moreover, there were
presentations of surveys of researches in the field (Grugnetti, Mariotti, Polo,
Vighi) and videos on exhibitions and didactic experiences on the theme
chosen (Vighi, Villani and Zuccheri).26
21 This work was gathered in a book (Arzarello et al. 1994) and was discussed in a
forum at PME 19 (Recife, Brasil), see Arzarello et al. (1995).
22 Some aspects of these contributions to the seminar can be found in Chiarugi et al.
(1990), Gallo (1994c, 1994d), Gallo et al. (1994), Menghini (1994a).
23 This theme was chosen according to the fact that the ICMI Study on Geometry had
been announced.
24 See Bartolini Bussi (1996), Bartolini Bussi & Pergola (1996), Gallo et al. (1991),
Gallo (1994a, 1994b, 1995a).
25 This is reported in Menghini (1992, 1994a), Maraschini & Menghini (1992). This
contribution highlighted the value of a teaching path aiming at theoretical thinking
through an operative genesis of geometrical concepts.
26 The video on the interesting lab-exhibition 'beyond the mirror" is enclosed in Zuccheri
(1996).
39
From all researchers' point of view, we must underline that though crossed by
various visions, the structure of the seminar became stronger and more
institutionalized in the Nineties. Within this structure, special commissions
were specifically created in order to document the Italian researches.
The first of these documentary studies, edited by M. Barra, M. Fasano, M.
Ferrari, F. Furinghetti, N. A. Malara, F. Speranza, was a booklet containing
the synthesis of the work done in the Eighties on psychology of maths
education, which was given to all participants at PME 16 (Assisi, 1991). To
create this booklet, the commission made a selection of all the abstracts they
had received. One of the criteria for selection was that the articles must have
been published on at least national scientific reviews.
The second study (Barra et al. 1992) was more complex. It concerned the
realization of a book for ICME 7, which was then introduced in a national
presentation (Quebec 1992). The first part of this book traced the main steps
and the relevant choices made for mathematics education in Italy from our
national Unification to now. It was divided into the following historical
periods: 1) from Unification to the first post-war period (1861-1921); the
fascist period (1922-1945);
3) from the second post-war period to the Nineties (1945-1992).27 The second
part of the book, realized with everybody's contribution, contained a selection
of papers appeared in Italy in the decade 1980-1990 in all research fields of
maths education.28
In 1994 a new committee was elected: N. A. Malara, M. Menghini and M.
Reggiani 1994/95). It was then, that a proposal of the previous committee
(see table 2) was accepted: themes and presenters would no longer be decided
autonomously by the committee, but rather chosen (still by the committee)
from among researchers' self-nominations, which however should be well
documented and substantiated. This committee also carried out and approved
a new formulation of the document of constitution of the national seminar.
This further innovation was an important moment for the growth of the
community: the scientific ripeness and autonomy achieved by many
researchers was now fully acknowledged, as well as the new spirit of cooperation introduced, respectful of all components and fully responsible for a
constitution project of an Italian scientific community in this field.
So, the following seminar sessions were chosen from among the proposals
presented. From 1993 to 1995, three sessions took place.
The first seminar of the three was devoted to representation in mathematics,
and it was realized by P. L. Ferrari, E. Lemut, M.A. Mariotti and A. Pesci,
27 The themes discussed concern school organization (syllabuses, textbooks, teachers
training), reviews and other publications, associations, mathematicians' contribution to
the improvement of maths education.
28 The selection was quite a demanding work, which took the committee long time for
reflection and the reading of many of the articles presented by the authors.
40
working together for the first time. After a common theoretical framing of
representation, each of the presenters analysed the theme from a specific
viewpoint, with precise reference to their studies. More precisely, Ferrari
analysed the role of figures in enacting problem solving strategies and their
"operatory state", Lemut studied the influence of representations in the
production of hypotheses for the solution of verbal problems by primary
school pupils, Mariotti discussed the role of drawing in its relation (or
conflict) with geometrical representation, Pesci analysed the role of graphical
mediator in mathematical visualization and learning, lingering in particular on
the role of tree graphs in the development of probabilistic reasoning at middle
school level.29
There was something new in this seminar: among the reactors there were,
beside a pedagogist, A. M. Aiello, the elementary school teacher F. Ferri and
the middle school teacher R. Iaderosa, both excellent researchers.
The second seminar concerned a new theme for the research tradition (which,
has we have seen, concentrated mainly on compulsory school), and rather upto-date: the problems about the mathematical knowledge of freshmen at the
faculty of mathematics, which can be framed in the more general social
problem of the passage from secondary school to university.
This research, which is wide and very well documented, was carried out by G.
Accascina and a group of secondary school teachers: P. Berneschi, S.
Bornoroni, M. De Vita, G. Della Rocca, G. Olivieri, G P. Parodi, F. Rohr. It
was based on a quite long questionnaire concerning basic mathematical
issues, which was aimed at studying the connection between the knowledge
presumed by secondary school teachers, the knowledge expected by
university professors and the students' actual knowledge. P. Boieri and C.
Fiori were invited to present similar researches carried out, respectively, at
the Polytechnic University of Turin and at the Faculty of Science and
Engineering of the University of Trieste.30
The third seminar concerned the researches on the learning of numbers in
elementary school. It was carried out by P. Boero together with E. Scali,
teacher-researcher, and by L. Cannizzaro with P. Crocini, teacher-researcher.
After a historical framing (Frege, Peano, Enriques as to the cardinal, ordinal
and measure aspects of numbers) and a cognitive, psycho-pedagogical
introduction (from Piaget to the more recent researches of constructivist
approach, of activity theory and reification), Cannizzaro and Crocini
presented their research on the spontaneous conceptions of numbers and
operations at the beginning of elementary school, whereas Boero and Scali
29 On the contributions presented, see Ferrari (1996), Lemut & Mariotti (1995), Mariotti
(1996), Dettori et al. (1996), Pesci (1994).
30
On the contributions presented, see Accascina et al. (1998), Boieri & Tabacco (1995), Boiti &
Fiori (1997).
41
analysed the long-term evolution of numbers mastery in the Genoa project
"Children, teachers, reality".31
It is in this period that the seminar accepted N. A. Malara's proposal to
publish a book for ICME 8 (Seville, 1996), containing essays on the
researches carried out in Italy in the previous 8 years. The book, called Italian
Research in Mathematics Education: 1988-1996, was edited by the
commission in office. It contained wide reports on the following themes:
arithmetic; algebra; intuitive geometry - rational geometry; logic; analysis;
probability, statistics and mathematics applied to other disciplines; computers
and mathematics; history and epistemology; problems; mathematics and
difficulties; theoretical models of teaching-learning processes; image,
conceptions and spreading of mathematics. It was created thanks to the
contribution of 34 researchers, working in small groups, and presented in
Seville by a delegation of authors.
The following committee was constituted by P. L. Ferrari, M. Reggiani and
R. Zan (1996/97). Choosing from among the proposals made, they organised
two seminars. The first was devoted to the impact of multimedial systems on
mathematics and presented by M. R. Bottino and G. P. Chiappini as well as
A. R. Scarafiotti and A. Giannetti. The second concerned the problems in the
passage from arithmetic to algebra in middle school and was presented by N.
A. Malara and the teachers-researchers L. Gherpelli, R. Iaderosa and G.
Navarra.
In the first seminar, after a theoretical framing about the kind of mediation
offered to the learning of mathematics by microworld-based systems, Bottino
and Chiappini presented the ARI-LAB system (created by them) and the
results of some experimentations,32 whereas Scarafiotti and Giannetti lingered
on the use of hypertexts in the didactic of mathematics, also producing
research results on this topic. The second seminar, focusing on the theorypraxis relationship, regarded a complex study on: a) changes of knowledge
and conceptions in teachers involved in an innovative project on the approach
to algebra, b) pupils' cognitive and metacognitive achievements.33
The commission elected in 1998/99 was constituted by L. Bazzini, L. G.
Chiappini and P.L. Ferrari . Two seminars were organized about coordinated
researches on the approach to theoretical thinking, carried out at different
school levels by F. Arzarello, M. G. Bartolini Bussi, P. Boero, M. Mariotti
and their staff. the first seminar, presented by Bartolini Bussi and Boero with
the contribution of R. Garuti and L. Parenti, was devoted specifically to
compulsory school and studied the processes of construction of the meanings
of statement, of theorem starting from real-life situations and, at a more
31. Reference to what exposed can be found in Boero (1990, 1994), Cannizzaro (1992, 1993),
Scali (1995).
32 See Bottino & Chiappini 1997a, 1997b.
33 See Malara (1996, 1997), Malara & Gherpelli (1996), Malara & Iaderosa (1998 and 1999).
42
advanced level, of theory through the mathematization of complex
situations.34 The second seminar, presented by F. Arzarello and M. A.
Mariotti with the contribution of F. Oliviero, D. Paola, O. Robutti, was
devoted to secondary school and dealt with the use of Cabri to formulate
conjectures in activities of geometrical exploration. In particular, it examined
the role of this environment, also with reference to other mediation tools such
as paper and pencil, in the passage from the experience phase to the
theoretical one on constructing statements to be demonstrated and on proving
them.35 The researches presented in both seminars were based on the
following theoretical constructs: a) experience field (introduced by Boero);
mathematical discussion (Bartolini Bussi), theorem (Mariotti et al.), cognitive
unit (Garuti et al.). In these seminars, the themes were presented from many
different points of view (didactic-methodological, cognitive, historicalepistemological, etc.), with the control of the different variables in each of the
points of view considered, as it appears in the files, which purposely contain
more details than the previous ones. These many aspects witness the growing
complexity of the trends of the most recent Italian research.
In particular, it must be underlined that for the first time, in the last seminar,
among the presenters there were young researchers and PhD students, who
worked with remarkable competence and autonomy.
It was then decided to publish another book (an ideal sequel to the previous
ones), this time for ICME 9 (Tokyo, August 2000), to illustrate the Italian
researches carried out in the second part of the Nineties and to start tracing
the history of the National Seminar (this paper is the result of such intention).
The community invited the committee in office to edit the book together with
Malara, since she had gained experience in the edition of the previous ones.
The committee which is currently in office (L. Bazzini, L. Cannizzaro and
G.P. Chiappini) was elected in December 1999 for the biennium 2000/01.
Since there are now many young researchers in the nuclei, the committee
approved the wish expressed by many participants that these young
researchers should organize the next seminar session (January 2001)
completely on their own, so as to promote their cooperation as well as their
individual growth. The theme chosen, "The difficulties in learning analysis"
in secondary school and at university, is based on their current experiences,
some of which are going on abroad.
This seminar shall mean an important generation renewal in the spirit of ideal
continuity.
34 See Bartolini Bussi (1998), Bartolini Bussi et al. (1999), Boero et al. (1996, 1997,
1998, 1999), Garuti et al. (1996, 1998), Parenti (1999?).
35 See Arzarello et al. (1998 and 1999a, 1999b, to appear), Mariotti & Maracci, Mariotti
et al. (1997).
43
SHORT REPORTS OF THE SESSIONSOF THE “ SEMINARIO NAZIONALE”
1st session (January - February 1987)
Theme: Measure in primary school36
Speakers: Maria G. Bartolini Bussi (Università di Modena e Reggio E.),
Paola Bandieri (Università di Modena e Reggio E.), the teachers: Cristina
Tioli, Annamaria Andreini, Fabiola Beretta (of the research group of the
Università di Modena).
Reactors: Claire Margolinas (Università di Parigi) and Giovanni Prodi
(Università di Pisa).
The speakers, even though they are aware that the concept of measure may be
investigated from different perspectives (for example, it plays a major role in
the approach to natural numbers, to rational-decimals and to reals), adopt a
geometrical standpoint, dealing with lengths, areas and volumes and
supporting the comparison to the approach to measure proper of experimental
science.
The seminar is organized in two parts, in the first theoretical aspects are dealt
with, in the second results of specific studies are presented; more precisely:
Theoretical aspects
Introduction and reference frame (Maria G. Bartolini Bussi);
The theory of quantities in geometry (Paola Bandieri)
Measure in experimental science (Cristina Tioli)
Specific studies
Projects and textbooks analysis (Maria G. Bartolini Bussi)
A study in a kindergarten (Maria G. Bartolini Bussi)
A workshop in primary school (C. Tioli)
A multidisciplinary study (A. Andreini, F. Beretta).
As this was the first session of the National Seminar, more than on specific
results, the presentation has focused on methodological aspects of the
different studies in order to start a debate on epistemological and
methodological foundations.
2nd session (July 1987)
Theme: Probability in Middle School37
Speakers: Angela Pesci (Università di Pavia), Maria Reggiani (Università di
Pavia), Carla Joo (teacher of the research group of the Università di Pavia)
36 Abstract by M.G. Bartolini Bussi.
Abstract by A. Pesci and M. Reggiani
37
44
Reactors: Mario Barra (Università di Roma), Elda Guala (Università di
Genova), Fortunato Pesarin (Dipartimento di Statistica, Università di
Padova).
In this session has been presented and discussed the part related to probability
of a study on the teaching of Probability and Statistics in Middle School,
carried out from 1979 to 1985 by the research group of the Università di
Pavia.
Organization of the Seminar:
1. Short presentation of the content matter of the project
Classical definition of probability as ratio proposed through the model of
the box, use of tree-graphs as a tool for the resolution of conditional
probability problems, some application to genetics, the law of large
numbers.
A characteristic of the project are remedial and improvement activities
related to other subjects usually dealt with at the same level, like fractions,
percentages, functions, equations, inequalities, literal calculus.
2. Motivation of some choices
Some choices related to contents and presentation methods, both with
reference to different conceptions of Probability (classical, subjective,
frequentist, ...) and to an axiomatic theoretical presentation have been
motivated through their insertion in a theoretical and epistemological
frame.
3. Methodology
The Project methodology has been discussed as concerns both the
planning stage with the teachers and the activities in the classes. Related to
this aspect the role of teacher as a coordinator of the works carried out by
students on individual or group tasks.
4. Presentation of some stages of the work in class.
5. Protocols related to single students' individual activities and have been
presented and discussed, dealing with evaluation and difficulties as well.
6. Comparison with other research lines on the same subject.
3rd session (December 1987)
Theme: Teaching and learning problems in Geometry. A preliminary study
for the construction of a didactical situation: reference system and space
geometry
Speaker: Maria Polo (Università di Cagliari).
Reactors: Maria G. Bartolini Bussi (Università di Modena), Vinicio Villani
(Università di Pisa)
The research under discussion, developed at the 'equipe de recherche en
didactique des mathématiques' of the University of Grenoble, belongs to the
area of studies on the problems posed by 'natural language' and 'scientific
45
language' in learning environments. It concerns a class situation in which the
use of natural language to express space geometry concepts and relationships.
More precisely, it is a didactical situation experimented in three primary
school classes, requiring two different stages of coding and decoding of
written messages with no pictures. In the first stage pupils can dispose of an
object made up of embeddable cube units that they are asked to describe
(without disassembling it) in a message in order to anable others to build it
again; in the second stage the goal is the reconstruction of the object, with the
availability of cube units, on the basis of a message produced in the first
stage.
From a methodological perspective, the study has developed into: a) a priori
analysis of the situation with the recognition of children knowledge and of the
strategies they possibly could enact; b) observation of the experimental
process, analysis and processing of results.
Organization of the Seminar:
1) Introduction, knowledge-oriented stage of the experience (reconstruction
of the experimental device through small group work), systematization
and first theoretical framing of the experience in class;
2) Theoretical frame of the research: references to other fields (psychology
and history of mathematics, ...), references to issues belonging to
mathematics education (space geometry teaching and learning, theory of
situations);
Description of the experimental device (the models of expected strategies, the
models of pupils' conceptions), analysis of some conjectures formulated in
connection to the goal of the study and their evaluation through the analysis
of protocols, the description and analysis of results.
4th session (May 1988)
Theme: Cognitive and metacognitive aspects of problem solving in primary
school.
Speaker: Ferdinando Arzarello (Università di Torino).
Reactors: Carlo Dapueto (Università di Genova), Claude Janvier (Cirade,
Canada)
The Seminar is divided into two parts. In the first one, after a survey of
related literature, a model for the analysis of problem solving activities is
presented which has been developed by the speaker and his team. Such model
is based on the assumption that the resolution of a problem requires: i) a lot of
structured knowledge on the field the problem deals with, ii) a system of
procedures to represent and trasform the problem, iii) a control system to
guide the selection of knowledge and procedures in order to find a solution;
from the integration of knowledge (as a corpus of ideas) and processes (both
cognitive and metacognitive) the pupil builds conceptual models
46
(i.e.cognitive structures) which allow him or her to succeed in problem
solving. Such model are affected by contextual, relational, lexical data, and
for this are unstable; they progressively become stable if the pupil can start
the control processes in order to integrate and discriminate them related to the
various problem situations. A conceptual model becomes stable when a good
balance is attained between the polarity pairs ‘semantics-syntax’, ‘natural
language-formalized language’ and the four connected basic fields (context,
vocabulary, algorithms, mathematical ideas).
The question dealt with is the study of how these factors affect the integration
and reinforcement process which determines the transition from unstable to
stable models.
Such investigation is carried out from different perspectives, according to the
situation and the contract (with analysis of processes vs. concepts, local vs.
global aspects, bright vs.poor problem solvers’ behaviors, attainment vs.
failure, individuals vs. groups).
The cognitive, metacognitive, epistemological questions arisen include: short
cuts in deduction and by-passes; integration between solution and
representation; instability of conceptual models; cognitive adjustements;
choice, control and application of unstable models; influences of contract on
the use of models; pupils’ beliefs; link between intuitive procedures and their
formal counterparts.
The second part is devoted to the experimental aspects of the study and to the
theoretical arrangement of results. At first the criteria by which pupils’
productions are analyzed to focus on cognitive adjustement processes are
presented. Such criteria consider conceptual models from two viewpoints: a)
in their relationship with mathematical knowledge as organized by the pupil;
b) in their organization and differentiation processes in relation with the
variety of problem situations involved in the problem field under
investigation, the text and the context.
Through the comparison of the detailed investigations carried out from the
two perspectives, pupils’ conceptual models in the fields of additive and
multiplicative problems are taken into account and a hierarchy of cognitive
models used is defined. Such hierarchies result largely consistent with other
well known ones, like Carpenter & Moser’s classification of additive
problems and Vergnaud’s classification of multiplicative ones (extending this
last one in various ways), and moreover, the theoretical frame developed
seems more flexible and befitting from an educational standpoint.
5th session (December 1988)
Theme: Introduction of basic ideas of computer science in primary and
middle school
Speaker: Margherita Fasano (Università di Roma “La Sapienza”)
47
Reactors: Fabrizio Luccio (Università di Pisa), Michele Pellerey (Pontificio
Ateneo Salesiano, Roma)
The seminar is divided into two parts. In the first one, after recognizing the
social value of the teaching of computer science, we deal with the problem of
what is to be introduced in primary and middle school and how to provide
basic instruction aven without the availability of computers. In our approach
computer science is regarded as a field linked to all the subject matters and
five 'organization concepts' are devised: information, databases, algorithms,
automata, systems of transformational rules, regarded as elements around
which it is possible to build a network of links related to other concepts, not
only from computer science. Focusing on and representing the relationships
among these elements and other ones taken into account successively, a
general conceptual map is built up, even if potentially improvable and
extendible, that make easier the global control of them and points out the
widespread role of computer science.
Some results are reported from psycholinguistics and evolutive and cognitive
psychology that sketch models of the organization and structure of the
knowledge stored in human memory, through hypoteses on the ways by
which data are gathered, processed, encoded, memorized. The fundamental
role played by language in the communication of information is stressed and
the contributes of computer science in the explicitation of thought processes,
in communication and in the support to symbolic thinking. An interpretation
of the official curricula for primary and middle school is provided, focusing
on both metodological suggestions and general purposes, and the contents
(not only mathematical ones), from the perspective of concepts and skills
proper of computer science education.
In the second part the pedagogical problem of the organization of teaching
sequences is dealt with. An interpretation of conceptual maps, previously
presented in a context of teaching planning based on contents, and a
taxonomy of general and specific goals is added (according to Metfessel &
Michael) about skills that are to be acquired.
Three schemes are presented that express the links between: a) basic goal organizing concepts; b) organizing concept - subject matter; c) goal organizing concept - subject matter.
(The specific-goal oriented taxonomy considers: prerequisites in terms of
‘know how to do’, knowledge in terms of organizing concepts of computer
science, logical skills, operational skills, attidudes, resources, applications).
6th session (May 1989)
Theme: Semantic fields in the teaching and learning of mathematics:
reflections on the conceptualization and linguistic mediation problems
related to curricular innovation experiences
48
Speaker: Paolo Boero (Università di Genova)
Reactors: F. Arzarello (Università di Torino), C. Dapueto (Università di
Genova).
The Seminar is devoted to a theoretical and critical revision of the Genova
group projects for teaching in primary and middle school, coordinated by the
speaker, focused on mathematization activities of reality and on the
construction of mathematical concepts and skills as tools for knowing real
world. A common feature of the projects is the fact that the work of
construction of such concepts and skills is carried out in different fields,
mostly through processes that aim at rationalizing pupils’ experience and
their knowledge of important aspects of their environment, arranged in
thematic areas that develop in a gradual and systemic way during long
periods. In the Seminar, at beginning, the motivations of the theme of
research are presented related to various aspects: i) the frame the group’s
research in locally included and the question of transferability; the status of
educational research and innovation in Italy; iii) the status of international
research and the question of the quality and coordination of italian presence.
We focus on methodology related to the data used (case-studies, (possibly
long-term) episodes of work in class, clues resulting from non systemic
observations by teachers or researchers in a number of classes, standard
assessment) and underline that such variety of data allow us to provide a
dynamical insight of the research developed and to corroborate results better
than with the use of standard assessment only.
Afterwards we go deep into the research with the presentation of the
theoretical definitions of ‘experience field’ and ‘semantic field’, introduced in
order to plan and analyze the educational re-contextualization of
mathematical knowledge with special care to the context linked to real world.
In particular the concept of semantic field, that is the idea around which our
research is organized, and is devised as a link between ‘experience fields’ and
‘conceptual fields’ (Vergnaud). Some problems of concept and procedure
construction linked with semantic field (cognitive functions of semantic
fields, the role of teachers, the management of work in class) are presented
and the question of linguistic aspects of the relationships between knowledge
of reality and teaching /learning of mathematics is dealt with, with some
effort in order to point out the conditions that make such relationships more
productive. This last point is treated more diffusely pointing out the role of
verbal language (at the levels of vocabulary, syntax, textual and about the
euristic or planning function, about the explicitation function and the
reflection function in connection to metacognitive aspects), of algebraic
language and its functions (stenographic, generalizing, transformational), of
iconic or, more generally, graphic language.
The research is framed in literature with reference to various settings: a)
mathematics education (the theory of didactical situations (Brousseau), the
49
recontextualization of mathematical knowledge, from a theoretical standpoint
(Chevallard) or more directly related to teaching engineering (Douady); b)
studies of ethnologic educational character (Bishop, Carraher, Nesher); c)
studies on the implications of constructivism on the construction of
mathematical concepts and skills in school environment (Kilpatrick, Sinclair,
Vergnaud, Cobb); d) studies in psychology of learning on the role of external
context in problem solving (Lesh), in the activation of the concepts of number
and operations (Gelman, Moser); in psycholinguistics (French, Nelson); on
conceptualization processes in a cognitivist paradigm (Nelson, Schank and
Abelson, Olson, Pontecorvo) and, mainly, in reference to Vygotskij's work
(influence of verbal language and teacher's mediation on the development of
mathematical thought).
The main statement of the study, supported by detailed references to teaching
experiments planned according the theoretical constructs presented above,
claims that the activities focused on the knowledge of real world, if guided
and linguistically mediated by the teacher in a suitable way, provide a number
of opportunities of a deep and effective construction of mathematical
competence at different levels (from the level of concepts and procedures that
directly take part in the knowledge of real world to the level of conceptual,
linguistic and metacognitive prerequisites necessary for 'internal'
mathematical activities on mathematical 'objects'). Some shortcomings of
teaching practices organized according to semantic fields as a possible
foundation for mathematics education and some critical points in the related
teaching planning are pointed out.
7th session (January 1990)
Theme: The interplay between images and concepts in geometrical reasoning
Speaker: Maria A. Mariotti (Università di Pisa)
Reactors: Ciro Ciliberto (Università di Roma “Tor Vergata”), Clotilde
Pontecorvo (Dipartimento di Psicologia, Università di Roma “La Sapienza”)
Guest reactors: Gabriele Di Stefano (Dipartimento di Psicologia, Università
di Padova), Pasquale Quattrocchi (Università di Modena), Carlo Scoppola
(Università di Trento)
The seminar deals with geometrical figures and their dynamics in the
development of geometrical thinking. Geometrical figures are referred to as
‘figural concepts’ (FC) for their double identity as mental products that
express concepts, and figural entities apt to represent spatial relationships
(Fischbein). The presentation starts from the consideration that a figure that
represents some geometrical problem may be mentally arranged in
completely different ways according to theory the problem is considered
within. For example, if one considers the problem “draw a circle and a point
P outside it, draw the tangents from P to the circle and name H, K the contact
50
points; prove that PH and PK are equal”, the treatment of the figure changes
if one regards the problems within euclidean geometry or transformation
geometry.
Therefore there are two control systems, one related to figural aspects, the
other to logical-conceptual ones, that regulate the dynamics of FC.
The study is based on the hypothesis that the congruence between the figural
and conceptual aspect is usually only partial and not always well balanced,
and even conflictual in some cases. In particular, the prevalence of a kind of
process on the other may provoke some fracture between the two aspects,
which leads to errors, difficulties, misconceptions.
Then a study is presented aimed at investigating on the dynamics of FC
related to a specific problem: the development of a polyhedra and,
conversely, their reconstruction, with no physical model available (in order to
promote a purely mental processing). The problem has been chosen for the
small amount of geometrical knowledge involved, because it requires good
skills of mental visualization and reasoning and because of the opportunity of
finding different versions with increasing degrees of complexity. The solids
taken into account are a cube, a right-angles parallelepiped, a regular
tetrahedron and a prism with triangular base. The research method consists of
interviews with a pre-arranged scheme of questions, based on the convinction
that just dialogue may show more clearly the interactions figural-conceptual
within pupils’ mind. A number of hypotheses, arisen from a previous
observational study, are tested like for example: i) in problem solving, the
partition of the problem into autonomous and meaningful sub-units by the
pupil; ii) in reconstruction problems, the correlation between the complexity
of the task of recognizing the pairs of sides that are to be matched and the
number of operations required. The influence of verbalization on the show of
the interactions figural-conceptual within the description of the resolution
process of a problem.
8th session (December 1991)
Theme: Educational research and teaching practice
Speakers: Ferdinando Arzarello (Università di Torino), Paolo Boero
(Università di Genova), Nicolina A. Malara (Università di Modena), Carmelo
Mammana (Università di Catania), Michele Pellerey (Pontificio Ateneo
Salesiano, Roma), G. Prodi (Univ. Pisa)., Benedetto Scimemi (Università di
Padova), Vinicio Villani (Università di Pisa). The teachers: M. Rocco (nucleo
di Trieste), C. Testa (nucleo di Torino), M. Trevisan (nucleo di Pavia)
Guest reactors: Cesare Parenti, Mario Gattullo (Università di Bologna)38
38 Parenti e Gattullo did not actually attend the seminar, the first for unknown reasons,
the second for an accident that brought him to death.
51
The seminar is organized in two panels, respectively devoted to: a) research
in mathematics education (Arzarello, Boero, Scimemi); interplay between
educational research and teaching practice (Malara, Mammana, Villani).
In the first panel Arzarello outlines research trends of mathematics education
in Italy from the sixties, within the frame of the cultural and social contexts
they have been influenced by. The exposition focuses on the increased
complexity and improved scientific level of recent research, that takes into
account more variables and their relationships in the observation of teaching
and learning processes. Boero presents a study of his on algebraic formalism
as a paradigm of educational research. Scimemi points out the instability and
the weakness of research and the need of contacts with international research,
in order to both pick up ideas and make italian studies better known.
In the second panel the following points are discussed:
1) The influence of educational research on educational practice.
2) How to realize an acceptable way to evaluate projects: the problem of
reproducibility
3) Relationships between educational research and in-service training
4) Diffusion of results and different kinds of publications
Mammana and Villani underline the difficult problem of the evaluation of
studies as research products, and are sceptical on the effectiveness of the
influence of research on school practice. Malara, with reference to her own
professional experience and to some international studies too, claims that it is
possible to realize innovation and affect everyday school practice if
researchers work at the side of teachers and with the aim of getting them
acquire new perspectives and awareness. She particularly stresses the
problem of reproducibility, linking it to the problem of the control of basic
variables of the process of teaching and learning (teachers’ culture and
conceptions, customs and contracts in the class, interaction dynamics, pupil’s
participation and his/her involvement in the process).
Pellerey presents a wide survey on different, sometimes opposite, research
models that are adopted in the international context of research. Rocco, Testa
and Trevisan speak of their position as teacher-researchers and its
implications, with referenceto their own experiences in their research groups.
9th session (December 1992)
Theme: Teaching and learning of algebra: state of things, methodologies of
research trendes and perspectives
Speakers: Ferdinando Arzarello (Università di Torino), Luciana Bazzini
(Università di Pavia), Giampaolo Chiappini (I.M.A.-C.N.R. - Genova), Elisa
Gallo (Università di Torino), Fulvia Furinghetti (Università di Genova),
Marta Menghini (Università di Roma).
52
Reactors: Carlo Bernardini (Università di Roma), Laura Toti Rigatelli
(Università di Siena).
Arzarello, Bazzini and Chiappini present a theoretical model on learning
processes in algebra. Such model is based on: i) the results of experimental
studies that ascribe to interpretations many of the difficulties in algebra
learning; ii) the basic ideas of Frege's semantics, and in particular the well
known distinction between sense (Sinn) and denotation (Bedeutung) of an
expression (Zeichen); iii) concepts belonging to traditional linguistics,
semiotics and modal logic. As regards algebraic expressions, the authors
distinguish between its denotation (i.e. the set of numbers represented by the
expression related to a given number system), its algebraic sense (i.e. the
explicitation of how what is denoted may result from the application of
computational rules) and contextualized sense (i.e. the sense that the
expression acquires within a given knowledge domain). Starting from such
distinction and through the detailed analysis of specific protocols of algebraic
problem solving activities they model algebra education as a game of
interpretations through different resolution worlds, where by 'resolution
world' it is meant a world where a pupil can produce an interpretant related to
a given problem situation, and which is defined by the mutual relationships
among the components it is characterized by (the kind of problem, the sign
system adopted, the features of social interactions, the knowledge of the
subject, his/her attitudes). The central aspect of this model is anyway the
dynamics of functional relationships that occur between senses and
denotations of algebraic expressions.
Gallo deals with the problem of the teaching of literal calculus and presents a
detailed analysis of the mental dynamics and the kind of control activate by
the pupil in the resolution process of algebraic syntax problem. Furinghetti
presents a report on difficulties on the learning of the concept of function,
including epistemological ones, with a survey of the more relevant
contributions on this issue that can be found in literature. Menghini39
presents a report pointing out what is left of literal calculus in the last three
years of high school. She undelines the importance, from an educational
perspective, of the distinction between arithmetic and symboplic algebra,
according to Peacock, and of the extension given to semantic aspects of
algebra related to modelling and interpretation in non-mathematical contexts.
She raises the problem of the availability of some form of semantic control
related to the 'form' of the algebraic expression, or to some perceptual skill.
She presents at last the outcomes of an experiment about such issue, wich
point out epistemological obstacles and difficulties such as the transition to
symbols, substitution and generalization.
39 Abstract by M.Menghini.
53
10th session (December 1993)
Theme: Geometry: epistemology, research methodologies, present trends. 40
Main speakers: Francesco Speranza (Università di Parma), Maria G. Bartolini
Bussi (Università di Modena e Reggio E.), Elisa Gallo (Università di Torino).
Other speakers: Lucia Grugnetti (Università di Parma), Maria Polo
(Università di Cagliari), Vinicio Villani (Università di Pisa), Luciana
Zuccheri (Università di Trieste).
Reactors: Michele Pellerey (Pontificio Ateneo Salesiano, Roma), Paolo Boero
(Università di Genova)
The complex contribute of F.Speranza, 'Some critical points from a
conceptual perspective regarding space' takes into account some conceptions
of space, regarded as implicit philosophies through which we look at it,
developing the issues by oppositions. He compares, in their historical
evolution, the conceptions of: inon-independent/ independent space;
absolute/relative
space;
homogeneous/non-homogeneous
space;
isotropic/non-isotropic space; bounded/unbounded space; finite/infinite
space; space as set of points/irreducible continuum; real space/form of our
perception; .... He widely treats the conceptions of space present in the
context of art and underlines how in such context some issues are dealt with
far before than in mathematics.
The contribute of Maria G. Bartolini Bussi 'Research methodologies: a casestudy' deals with the methodology of research that has been realized in the
group guided by her, with various research projects involving various school
levels:
ƒ Teaching/Learning of Mathematics in Kindergarten
ƒ Mathematical Discussion in Primary School (Perspective Representation
of the Visible World)
ƒ Mathematical Instruments in High School (Curves and Transformations)
In the presentation the following issues are discussed:
1) Progressive and continuous construction of a theoretical reference frame
2) The relationship between the theoretical frame and the problems as they
are dealt with
3) The relationship between the theoretical frame, the problems and some
research results.
The contribute of E.Gallo 'Modelling and Geometry at the age of 14-16'
includes two parts: in the first some studies of the Group's are presented
classified into two themes on 'Geometry, perception and language' and
'Actualization and evolution of models in geometry problem solvingthrough
40 The abstracts of the contributes of Bartolini Bussi, Gallo e Menghini are by the
authors.
54
the dynamics of control; in the second such studies are analyzed in order to
point out their type (phenomenological, qualitative, diagnostic) and, mostly,
their methodology (theoretical reference frame, compresence and
determination of didactical variables, evolution of observation methods).
The contribute of M.Menghini regards theoretical studies that may be
classified as historical-critical. Her purpose is to outline a clear survey of the
various positions concerning the teaching of geometry, related to both
educational goals and the philosophies underlying the various choices. Three
events are stressed which have strongly affected contemporary teaching:
1) The re-introduction in 1860 of Euclid's Elements as a textbook and the
progressive revisions leading to Hilbert 's 'Grundlagen der Geometrie';
2) Klein's Erlanger Program, Enriques' psychology-driven revision, Libois'
adjustements from a curricular and methodologic perspective;
3) Bourbakist revolution and the 'total' axiomatization of geometries.
The presentation focuses on: hypotetical-deductive structure and its
relationship with geometrical intuition; the foundation of geometrical figures
equality and the related function of 'motions' or 'transformations'; the purity of
geometry compared to arithmetic and algebra.
The contributes of L.Grugnetti, M.A.Mariotti, M.Polo and P.Vighi are some
surveys of recent studies in geometry published in international journals
internazionali (Educational Studies in Mathematics, For the Learning,
JERME, Journal of Mathematical Behaviour, Recherches in didactique de las
mathematiques,…). Further contributes are given by P. Vighi, V. Villani e L.
Zuccheri regarding the presentation of educational videos in geometry.
11th session (January 1995)
Theme: Representations in Mathematics and teaching-learning processes
Speakers: Pier Luigi Ferrari (Università del Piemonte Orientale), Enrica
Lemut (I.M.A.-C.N.R.-Genova), Maria A. Mariotti (Università di Pisa),
Angela Pesci (Università di Pavia)
Reactors: Aiello (Istituto di Psicologia, Università di Roma “La Sapienza”),
Franca Ferri (primary school teacher in Modena, researcher of the group
directed by M.G. Bartolini Bussi), Rosa Iaderosa (middle school teacher in
Cesano Boscone-Milano, researcher of the group directed by N.A. Malara)
The seminar starts from a theoretical analysis on the issue 'Representation in
Mathematics', with the presentation of some significant contributes from the
field of psychology and the comparison of the western approach (from Piaget
and Bruner to some recent contributes from the information processing
framework) to the Soviet one, focusing on Vygotskij's thought.
Then the speakers deal with representation in mathematics, presenting a
sequence of studies through which they point out:
55
a) conflicts and deadlocks caused by the lack of integration of more
representation systems and of mastery of italian language as well;
b) advantages and limits of various representation systems (verbal-arithmetic,
algebraic, technology-based, ...) in the solution of the same algebraic
problem;
c) the pertinence, from a logical standpoint, of geometry proofs that involve
figures;
d) the mix of representations, old and new, used in the construction of the
concept of proportionality in a sparse way;
e) the problem of using concrete models in geometrical education.
Some theoretical contributes on representation in mathematics from various
authors (Vergnaud, Fischbein, Dorfler, Goldin, Duval) are presented as well.
In the second part each of the speakers presents one specific study of his or
hers. P.L. Ferrari exposes a study related to geometry problems at the age of
8-10, where he investigates the operational status of the figure in the
resolution process. E. Lemut deals with the interplay between the productions
of hypotheses and of representations, at the level of primary school.
M.A.Mariotti investigates the role of the drawing in the modelling of real
phenomena, like sunshadows, analyzing the various ways of using it (from
prevision-oriented model to tool for reationalization). A.Pesci presents a
study on the use of tree-graphs in the solution of of composite probability
problems, comparing their role in different problem situations and pointing
out that it may be managed more easily if the temporal structure of the
problem situation is congruent with the temporal structure of the graph
generation.
12th session (October 1995)
Theme: Bridging problems in mathematics between high school and
university
Speakers: Giuseppe Accascina (Università di Roma “la Sapienza”) teacherresearchers: P. Berneschi, S. Bornoroni, M. De Vita, G. Della Rocca, G.
Olivieri, G.P. Parodi, F. Rohr.
Reactors: Pier Luigi Ferrari (Università del Piemonte Orientale), Carlo
Marchini (Università di Parma).
A study on mathematical knowledge of Mathematics and Engineering
freshman students is presented which is based on the conjecture that the
major difficulties of freshman students depend on the gap between their real
preparation, which is different from the preparation presumed by their high
school teachers, and their virtual preparation, i.e. the preparation presumed by
university teachers.
The research is based on a questionnaire, made up by 32 items (mostly
multiple-answer tests), on the following subjects: number sets and elements
56
of logic, algebra, plane euclidean geometry, transformation geometry,
exponential and logarithmic functions, trigonometric functions, approximate
solutions. Such subjects are mostly common to mathematics curricula of
various kinds of school.
In the study, which is very detailed, can be found:
a) a comparison between high school teachers' and university teachers'
conceptions about students' competence with reference to the items of the
questionnaire;
b) a quantitative and qualitative analysis of students' answers related to
various issues and taking into account:
i) the high schools students come from (schools with different mathematics
curricula);
ii) the degree course they have chosen (mathematics or engineering).
As regards a), the study points out a clear gap between high school and
university teachers' conceptions (e.g.: the basic elements of plane geometry is
attained in the opinion of the first ones but not in that of the second ones);
still, there is a certain agreement as far as algebraic competence is concerned.
A common feature of the answers of the teachers of both levels is their
reference to a well fixed model of student, generally a student from the 'Liceo
Scientifico', which shows some lack of awareness about people attending
university mathematics courses, and is a serious problem on the side of
university. If teachers' expectation are compared to actual data, i.e. to the
answers effectively given by students, it seems clear that teachers of all kind,
but mostly university ones, use to overrate students knowledge, that appear to
be very poor from the results of the study.
From the comparison between Mathematics and Engineering students, the
second one appear to be more clever, which could depend upon the different
schools from which the two samples come. A comparison restricted to the
students coming from just one kind of school shows a substantial parity. This
provides a pessimistic insight on the cultural quality of the prospective
mathematics teachers. It is argued that to reduce these effects varied and
student-oriented remedial courses should be offered, based on a detailed
assessment in order to make each student aware of his or her own gaps. As a
conclusion, it is argued that some remedial activities usually proposed by
universities (like bridging courses) could even have opposite effect, just for
the lack of control on their actual results. The study is equipped with a wide
survey of analogous studies in Italy and abroad.
13th session (January 1996)
Theme: The learning of number in primary school
57
Speakers: Paolo Boero (Università di Genova) and Ezio Scali primary school
researcher-teacher of his group, Lucilla Cannizzaro (Università di Roma “La
Sapienza”) and P. Crocini researcher-teacher of her group.
Reactors: Ferdinando Arzarello (Università di Torino), Rosetta Zan
(Università di Pisa)
The seminar is divided into two parts. In the first one the theoretical reference
frame of the specific studies dealt with in the second part is presented. Such
frame takes into account various contributions from:
a) history and epistemology of mathematics (Frege, Peano, Enriques for the
cardinal, ordinal, measure aspects of number);
b) psycho-pedagogical and cognitive research (from Piaget to the most recent
studies within radical constructivism, activity teory or the process-object
paradigm);
c) psychological studies linked to neuro-physiological ones; statisticdiagnostic studies on the mastery of number and of the knowledge of
number facts (Hart);
d) didactical studies on the teaching and learning of numbers, and in
particular those of the Brousseau's group (for their extension - from
enumeration to rationals - and paradigmaticality in the sense of 'didactic
research'), of the Dutch school (Streefland, Treffers, de Lange) and of the
Padova group, coordinated by C. Bonotto.
In the second part two studies are presented, coordinated by L. Cannizzaro
and P. Boero respectively. The first one, presented by P. Crocini, has been
realized in two times, based on pupils' protocols, records of interviews and
teacher's notes. It deals with the investigation of the competences constructed
by the child in his or her social interactions with adults, peers and the
environment, regarding numbers (counting, quantifying, reading and writing
numbers) and the operations (representing, taking note, computing). The
second deals with the investigation of the evolution of children's arithmetic
competences related to the problem situations, the contexts proposed and the
teacher's mediations in the frame of the Genova Project 'Children, Teachers,
Reality'. More specifically, the study involves processes of generation,
differentiation and meaning rupture of number related to the problem
situation proposed. In particular the focus is on:
i) the meaning 'value' of number in the experience field 'money and prices', as
a rupture of cardinality and in the perspective of the approach to positional
writing;
ii) the 'remainder problems' in economy, the mastery of meanings of
subtraction and the development of mental processes related to problem
solving;
iii) the additive decomposition processes, the development of mental
processes related to problem solving and the generation of the meanings of
58
the operations of addition and subtraction and their properties as 'theorems
in action'.
The methodology adopted is 'based on temporal sections' of the long-term
teaching and learning processes and 'based on the child' and, as the previous
one, uses: written protocols, records of interactions with the teachers and
teachers' notes.
14th session (December 1996)
Theme: Microworlds, hypertexts and communication systems in mathematics
education 41
Speakers: Rosa Maria Bottino e Giampaolo Chiappini (I.M.A.-C.N.R,
Genova); Anna Rosa Scarafiotti e Annarosa Giannetti (Politecnico di Torino)
Reactors: Gianna Gazzaniga (IAN-CNR, Pavia), Maria A. Mariotti
(Università di Pisa)
The contribute of Rosa Maria Bottino e Giampaolo Chiappini focuses on two
issues:
1) The nature of the mediation provided by microworld-based systems: state
of things and theoretical reference frame.
2) Analysis of the nature of the mediation provided by ARI-LAB system:
presentation of the experiments performed and open problems.
The examination of literature related to information systems for mathematics
learning has provided a key to recognize a unifying insight on the role of the
mediation offered by computers in the various experiences carried out by a
number of authors. Then the theoretical reference frame adopted by the two
speakers to account for the nature of the mediation provided by the
interaction with a microworld within the context of use of the system is
presented. Based on this frame the results of their research related to the
planning, implementation, experimentation and evaluation of a system based
on a number of microworlds and oriented to arithmetic problem solving
(ARI-LAB)
The contribution of Anna Scarafiotti and Annarosa Giannetti focuses on the
following two themes:
1) The myth of the frame. Knowledge and communication; contributes from
cognitive sciences and hypertext -based techiques.
2) Hypertexts realized for mathematics education: research presuppositions,
development perspectives, example of use.
Their presentation focuses on the issues concerning the interplay between
knowledge and communication in mathematics education and more in
particular on the sense construction related to hypertext-based methods. After
a theoretical framing of the issue, some features of hypertext-based methods
41
Abstract by G. Chiappini
59
are analyzed that seem most interesting and effective as far as teaching
practice is concerned. A short history of the studies carried out by the two
researchers which stresses the actual issues dealt with, including the use of
CABRI within a hypertext and the realization of applications of hypertexts to
teaching at the end of high school and at the beginning of university,
equipped with data gathered from experimental observations.
15th session (December 1997)
Theme: The problem of the transition arithmetic-algebra in middle school
Speakers: Nicolina A. Malara and the researcher-teachers Loredana Gherpelli,
Rosa Iaderosa and Giancarlo Navarra.
Reactors: Giampaolo Chiappini (IMA – CNR, GENOVA) and Elisa Gallo
(Università di Torino).
Guest: Luis Rico (Università di Granada)
The seminar is devoted to studies and experimental research carried out by
the group and focuses on the relationship theory/practice; it is a study based
on the observation of the development of a complex process that takes into
account:
a) teachers' conceptions and their slow evolution towards new forms of
awareness by means of the reading of and discussion on results of
international research;
b) the influence on educational and cultural options in the class;
c) students' development of skills and attitudes that are usually regarded as
'advanced';
d) the change in the conceptions and behaviors of the teachers involved.
The seminar is organized as follows.
1) Presentation and theoretical frame (N.A. Malara)
2) Survey of the studies carried out in international research, with special
reference on those regarding learning difficulties (N.A. Malara, G.
Navarra)
3) Experimental evidence from the observation of the same class during the
whole three years of middle school, with special reference to students'
productions related to the resolution of word problems and reasoning and
proving in arithmetic setting (L. Gherpelli, N.A. Malara).
4) Syntactical and structural aspects in the transition arithmetic-algebra (R.
Iaderosa, N.A. Malara).
Among the experimental studies are worth to mention (because of the
originality of the setting) those on structural analogies, which is a subject
included in official curricula but often misunderstood even by textbooks with
a wide circulation and usually overlooked in teaching because of the lack of
knowledge or of experience by teachers.
60
There is also a contribution of L.Rico (of the University of Granada) on some
studies carried out in his group on algebraic problem solving and the
classification of resolution processes.
16th session
(December 1998)
Theme: The first approach to theorems in primary and middle school.42
Speakers: Maria G. Bartolini Bussi (Università di Modena e Reggio E.),
Paolo Boero (Università di Genova), Laura Parenti (Università di Genova),
Rossella Garuti (middle school teacher at Carpi-Modena, researcher and
collaborator of Bartolini Bussi and Boero).
Reactors: Gilbert Arsac (Université de Lyon, France), Mario Barra
(Università di Roma “la Sapienza”), Maria Polo (Università di Cagliari).
Organization of the seminar:
Theoretical frame (Maria G. Bartolini Bussi, Paolo Boero)
The theoretical frame of the complex of studies is presented in its historicalepistemological, cognitive and educational components, all framed in related
literature. In the particular case of theorems in primary and middle school the
presentation focuses on:
A) Pupils' behaviors and difficulties in the customary approach to theorems
(cognitice and didactic analysis)
B) Nature of the object of teaching (historical and epistemological analysis);
such analysis should consider both the historical evolution of the ideas of
‘theorem’ and of ‘mathematical theory’ from Euclid to Hilbert up to
nowadays, and the invariants in such evolution (as they could play a
central role in didactical transposition);
C) nature of mathematical activity related to theorems (cognitive and
historical and epistemological analysis); such analysis seems extremely
important as it regards the theorem producing process and helps to point
out the difference between the production process and its final output.
D) recognition of the conditions that could put within the reach of pupils the
mathematical activity regarding theorems (cognitive analysis, in the sense
of individual and social cognition, and didactical analysis). The conditions
to be taken into account regard pupils’ background, the choice of the
themes on which they will work in their approach to theorems and, within
the themes, the choice of the tasks and their management by the teacher (in
particular as regards cultural mediation and discussion orchestration.)
The theoretical frame worked out by the group made up by Bartolini Bussi,
Boero and Mariotti and afterwardsdeveloped by Arzarello as well is based on
four constructs: Experience Field (Boero), Mathematical Discussion
42
Abstract by M.G. Bartolini Bussi.
61
(Bartolini Bussi), Theorem (Mariotti et al.), Cognitive Unity (Garuti et al.).
The general presentation of the theoretical frame has been followed by the
presentation of specific studies:
• Cognitive Unity in Theorems (Garuti)
• Construction problems in grades 5-8 (Maria G. Bartolini Bussi)
• From dynamical exploration to theory and theorems (Laura Parenti et al.)
17th session (December 1999)
Theme: Introducing students to theoretical thought: investigation on some
mediators 43
Speakers: F.Arzarello (Università di Torino), M.A. Mariotti (Università di
Pisa), F. Olivero (University of Bristol), D. Paola (high school teacher), O.
Robutti (Università di Torino)
Reactors: Claudio Bernardi (Università di Roma I ‘La Sapienza’, Angela
Pesci (Università di Pavia), Bernard Capponi (Université de Grenoble).
In the studies presented here classroom activities are both a means and an
outcome of the increasing knowledge of teaching and learning processes.
They show how students, at the age of 14-18, have succeeded in the
construction of theoretical meaning for geometry problems, producing
geometry theorems under various forms (constructions; proofs) mainly (but
not only) in dynamical geometry environments (Cabri Géomètre).
The reference frame includes the following elements of theoretical analysis:
experience fields (Boero); mathematical discussion (Bartolini Bussi);
semiotic mediation (Vygotskij); the idea of theoretical knowledge, with
particular regard to the status of theorems in the frame of mathematical
knowledge (Arzarello, Boero, Mariotti); the idea of cognitive unity (Boero,
Mariotti).
The experience field involved is given by the constructions and explorations
in Cabri environment. In a long term process the experience field evolves
through social activities focused on the resolution of open problems and
mathematical discussion with the presence of both the voices of practice and
theory.
Practice refers to students' drawing experiences, recalled by:
i) 'concrete objects' in paper-and-pencil environment (drawings, sketches,
instruments);
ii) 'computational objects' like Cabri figures or commands;
The theory is referred to geometrical objects, their properties, the logical
relationships among them and is recalled by:
i) dynamical processes perceptible on the screen;
43
62
Abstract by M.A. Mariotti & O. Robutti
ii) commands available on Cabri menu.
The interaction between the two voices is induced by activities in both paperand-pencil environment (the constructions with ruler and compasses) and
Cabri environment (constructions and dragging).
The interactions occurs if it is adequately supported by the teacher (whose
role is crucial), who uses the software to lead the students to the construction
of the theoretical meanings involved in the experience field.
The external context is given by:
i) concrete objects (drawings, sketches, ...) and artifacts (geometrical
instruments) in paper-and-pencil environment;
ii) computational objects and artifacts (Cabri-figures, commands, dragging)
iii) signs, including gestures.
The reference culture is given by:
i) classical geometry (which is expicit in Cabri commands);
ii) analysis/syntesis (which is implicit in Cabri dynamical processes);
iii) algebraic varieties theory (which is implicit in the software).
The internal context of the student is analyzed starting from the dialectics
intuitive/deductive geometry, which is present in a number of analyses on the
status of students' geometrical knowledge. Such model has been refined,
however, by the introduction of subjet's temporal dynamics, as regulators of
cognitive and didactic dynamics. The teaching of proof in geometry, in Cabri
environment points out a serious gap between the empirical and intuitive
aspects (that are supported by Cabri) and theoretical ones, that on the contrary
are not available in the software. Actually the a priori analysis shows that the
dialectics artifact/instrument, that is started by the use of Cabri in class, does
not naturally lead the students to theoretical knowledge involved in proof.
Theoretical knowledge involved in proof includes very subtle notions:
typically the status of geometrical objects, of their properties, of the logical
relationships among them, that are not explicitly embodied in Cabri. The
'didactical engineering' that seems more suitable to deal with this kind of
situations is not constructivist only but is based on the concepts of semioti
mediation and mathematical discussion and requires a strong intervention by
the teacher.
The function of the instrument grows more complex: the instrument is not
only used by a user, who, as such, in its interaction with it, develops potential
use schemata and supports the construction of knowledge related to such
schemata. The instrument is inserted between who is learning and who is
teaching and is used by the teacher to guide
Individual evolution processes within the community of class.
The semiotic mediation process centered on one instrument is developed on
two levels:
• the student uses the instrument according to schemata aimed at fulfilling
the given task;
63
• the teacher uses the instrument according to schemata aimed at fulfilling
his or her educational purposes (for example, strategies like the forecast
game with Cabri facility 'History', or other).
The construction of meaning takes place in the dialectics between the two
levels of activity/action.
So the instrument is subject to a double use, in relation to which plays its role
of mediator of meanings. From the learner's side it is used as an instrument to
carry out specific actions aimed at the fulfilment of a given activity; from the
teacher's side it is used to guide the student towards the construction of
meanings. In other words, if we think of the computer, it may take part to the
activities in different ways, according to time and actors.
• As an instrument (artifact used according specific use schemata) to carry
out a task; in this case the knowledge embodied in it may remain totally
inaccessible to the student user. The instrument is used to strengthen the
subject's available skills (e.g.: finding the limit of a function in Derive
environment by means of the command 'lim'; drawing a bisector in Cabri
environment by means of the command 'bisector') and develops use
schemata.
• As an instrument for semiotic mediation used by the teacher to realize her
or his communication strategies, aimed at the development of a specific
piece of meaning, related to the mathematical content involved in
teaching.
Mathematical knowledge, i.e. the knowledge embodied by the instrument, is
made accessible by its use, but it could become explicit only after specific
actitivies focused on the evolution/construction of mathematically acceptable
meanings. Meanings are based of phenomenological experiece (the user's
action and the feedback of the environment), but their evolution is achieved
by means of social constructions in the class, under the guide of the teacher.
The teaching project resulting from this approach consists in the
transformation of the two terms (activities with the artifact and theoretical
knowledge) into voices of a mathematical discussion: the discussion as a
'polyphony of voices' (Bartolini Bussi) will result after the introduction of the
voices related to the practice with Cabri and the voice of theory, progressively
built up in class and introduced as 'knowledge'.
In reference to the idea of reference field (Boero), the external context
includes objects (concrete ones like the drawings on paper or computational
ones like Cabri figures) and the operations on them (constructions, dragging,
...). Such objects recall the the voice of practice and of theory, and the
dialectics between them, guided by the teacher, marks the beginning of an
evolution process, where the use of the instrument, at first outwards-oriented,
becomes internal control. The internalization process (Vygotskij) of the ways
of use of the artifact, characterizes the evolution of the internal context of the
student.
64
The experiences carried out in these last years show the validity of a model
formulated this way; the detailed analysis of the protocols produced by the
students in the contect of the experimentations has allowed a detailed analysis
of the internalization process pointing out specific elements that take part. In
particular, the production of signs (within or without Cabri environment) by
the student that allow the transition from outside to inside.
The model described is consistent with the analysis, by other authors, of
lower age level (see Boero, Parenti, Bartolini Bussi). Moreover, it makes
deeper the analysis of the polarity intuitive/deductive developed by Arsac and
Laborde in terms of external context and reference culture; the explicit
introduction of the idea of semiotic mediation, of the dialectics of voices (of
practice and theory) and of internal context (temporal dynamics) seems to us
to work in order to give account in a more complete way of the evolution of
the experience field of constructions and explorations with Cabri.
Acknowledgments
We are grateful to our colleagues: M.G. Bartolini Bussi, G. Chiappini, E.
Gallo, M.A. Mariotti, M. Menghini, M. Reggiani, O. Robutti for their
cooperation in the preparation of the reports related to the sessions of the
Seminar they were involved.
We would like to thank C.Bernardi too for the abstract of his reaction at the
last session of the Seminar, which could not be inserted for the general lack of
materials related to the other reactions.
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DETTORI, G, GARUTI, R., LEMUT, E., MARIOTTI, M.A.: 1996, Evolution of
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GALLO E.: 1994a, Elaboration of Models for Problem Resolution in Interaction
with 14/15-year-old Pupils, proc. Second Italian-German Bilateral Symposium
on Didactics of Mathematics, Osnabrück 1992, IDM Band 39, 289-302
67
GALLO E., 1994b, Le figure, queste sconosciute: come manipolarle, disegnarle,
immaginarle per conoscerle meglio, L'apprendimento della matematica: dalla
ricerca teorica alla pratica didattica, Atti Incontri con la Matematica n. 8,
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1992, Rendiconti Seminario Matematico Univ. Pol. Torino, 52-3
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proc.CIEAEM 46, Tolosa 1994, 256-263
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situation de résolution de problèmes: contrôle descendant et ascendant, proc.
CIEAEM 41, Bruxelles 1989
GALLO E., BATTÙ M., TESTA C.: 1994, Evaluation et contrôle dans le calcul
littéral comme problème, proc. CIEAEM 45, Cagliari 1993,
GARUTI, R;, BOERO, P., LEMUT, E., MARIOTTI, M.: 1996, Challenging the
Traditional School Approach to Teorems: a Hypothesis about the Cognitive
Unity of Theorems, Proc. PME 20, Valencia, Spain, vol. 2, 113-120
GARUTI, R., BOERO, P. & LEMUT, E.: 1998, Cognitive Unity of Theorems and
Difficulty of Proof, Proc. PME 22, Stellenbosch, vol. 2, 345-353
LEMUT, E., MARIOTTI, M.A.: 1995, Pictures and Picturing in Elementary Problem
Solving, Proc. European Research Conference on Psychology in Mathematics
Education, Osnabruck, 42-45
MALARA, N.A., 1992, Ricerca Didattica ed Insegnamento, L’Insegnamento della
Matematica e delle Scienze integrate., vol. 15B, n. 2, 107-136
MALARA N.A.: 1996, Algebraic Thinking: How Can be promoted in Compulsory
Schooling Whilst Limiting Its Difficulty?, L’Educazione Matematica, XVII,
serie V, vol. 1, 80-99
MALARA N.A.: 1997, Problemi nel passaggio Aritmetica-Algebra, La Matematica
e la sua Didattica, vol. 2, 176-186
MALARA, N.A.: 1998, Didattica della Matematica: il caso italiano, to appear in
Hollenstein, A. & Grunder, H.U. (eds), proc. International Colloquium
"Fachdidaktik als Wissenschaft und Forschungsfeld in der Schweiz", (18-23
Ottobre 1998, Monte Verità, Ascona, Svizzera)
MALARA, N.A.: 1999, An Aspect of a Long Term Research on Algebra: the
Solution of Verbal Problems, proc. PME 23 , Haifa, Israele, vol. 3, 257-264
MALARA, N.A.: 2000, Francesco Speranza as Mathematics Educator: Values and
Cultural Choices in this book
MALARA N.A. MENGHINI M, REGGIANI M. (eds): 1996, Italian Research in
Mathematics Education: 1988-1995, Litoflash, Roma
MALARA N.A., GHERPELLI L.: 1997, Argumentation and proof in Arithmetics:
some results of a long lasting research, L'Educazione Matematica, anno XVIII,
serie V, vol. 2, n. 2, 82-102
MALARA N.A., IADEROSA R.: 1998, The Interweaving of Arithmetic and Algebra:
Some Questions About Syntactic, Relational and Structural Aspects and their
Teaching and Learning, to appear in proc. CERME 1
MALARA N.A., IADEROSA, R.: 1999, Theory and Practice: a case of fruitful
relationship for the Renewal of the Teaching and Learning of Algebra, in
JAQUET F. (ed) Proc CIEAEM 50 -Relationship between Classroom Practice
and Research in Mathematics Education, 38-54
MARASCHINI, M. MENGHINI, W.: 1992, Il metodo euclideo nell'insegnamento
della geometria, L'Educazione Matematica, XIII, 3, 161-180.
MARCHINI, C., 2000, The Philosophy of Mathematics according to Francesco
Speranza, in this book
MARIOTTI, M.A., 1989, Mental Images: Some Problems Related to the
Development of Solids, proc. PME 13, Paris, France, vol. 2, 258-265
68
MARIOTTI, M.A., 1996, Reasoning Geometrically trough the Drawning Activity,
proc. PME 20, Valencia, Spain, vol. 3, 329-336
MARIOTTI, M.A., SAINATI NELLO, M., SCIOLIS MARINO, M.:1987, L’immagine
mentale per la formazione dei concetti geometrici, Insegnamento della
Matematica e delle Scienze Integrate, vol. 10, n. 5A, 458-465
MARIOTTI , M. A., BARTOLINI BUSSI, M. G., BOERO, P., FERRI, F., GARUTI,
R.: 1997, Approaching Geometry Theorems in Contexts: From History and
Epistemology to Cognition, in proc. PME 21, Lahti, Finland , vol. 1, 180-195
MARIOTTI , M. A., BARTOLINI BUSSI, M. G.: 1998, From Drawing to
Construction: Teacher’s Mediation within the Cabri Environment, proc. PME
22, Stellenbosh, South Africa,vol. 3, 247-254
MARIOTTI , M. A., MARACCI, M.: 1999, Conjecturing and Proving in problem
solving situation, proc. PME 23, Haifa, Israel, vol. 3, 273-280
MENGHINI, M.: 1992, Piano affine e costruttivismo, La Matematica la sua
Didattica, 4, 5-13.
MENGHINI, M.: 1994a, Die euklidische Methode im italienischen
Geometrieunterricht seit 1867, in "Der Wandel im Lehren und Lernen von
Mathematik und Naturwissenschaften", Band I: Mathematik, Schriftenreihe der
Pädagogischen Hochschule Heidelberg, Deutscher Studien Verlag, Weinheim,
138-151.
MENGHINI, M.: 1994b, The Form in Algebra: reflecting with Peacock, on Upper
Secondary School Teaching, For the Learning of Mathematics, vol. 14, n. 3, 914.
NAVARRA, G., DE PLANO, S.: 1992, Gli insegnanti ricercatori in didattica della
Matematica: alcune note informative su un isola forse poco conosciuta
dell'oceano scuola, Insegnamento della Matematica e delle Scienze Integrate
vol.14, n.8, 791-794
POLO, M.: 1988, Sistema di riferimento e geometria nello spazio, analisi di
comportamenti spontanei di bambini di 8/9 anni, L’Insegnamento della
Matematica e delle Scienze Integrate, vol. 9A/B, 715-740
POLO, M.: 1989, Descrizione e ricostruzione di oggetti solidi: un’esperienza in terza
e quarta elementare, L’Insegnamento della Matematica e delle Scienze Integrate
vol. 12A, 1079-1092
PESCI , A:, 1994, Tree Graphs: Visual Aids in Casual Compound Events, proc.
PME 18, Lisbon, Portugal, vol. 4, 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. REGGIANI, M.: 1987, Statistica e Probabilità nella scuola media
inferiore: una proposta didattica, TID-CNR project, IDM series, vol. 1
PRODI, G.: 1975/77, Matematica come Scoperta, D’Anna, Firenze
PRODI, G.: 1992, Ricerca in Didattica della Matematica, Notiziario dell'Unione
Matematica Italiana 19 (1-2), 146-150.
SCALI, E.: 1995, Expériences des enfants dans le demaine economique et
construction des compétences de base en arithmetique, proc. CIEAEM 47,
Berlin, 409-413
SPERANZA, F.: 1994 Alcuni nodi concettuali a proposito dello spazio,
L’Educazione Matematica, anno XV, serie IV, vol. 1, n. 2, 95-115
SPERANZA, F., ROSSI DALL’ACQUA, A., 1971/174, Matematica, Zanichelli,
Bologna
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Matematica e delle Scienze Integrate, vol. 10, 242-258
VIOLA, T.: Sull’insegnamento delle materie scientifiche nella scuola media unica,
Periodico di Matematiche, serie IV, XLIII, 4-83
ZUCCHERI L.: 1996, La mostra-laboratorio "Oltre lo specchio”: note per gli
animatori e video cassetta illustrativa, DSM, Universita' di Trieste, Quaderno n.32.
69
70
The “Seminario Nazionale”: 1987-1999 – a synthesis
National
Committee
Arzarello,
Boero, Gallo
Barra, Boero,
Mariotti
Speakers
Bartolini Bussi, Bandieri,
Tioli, Andreini, Beretta
Measure in primary school
Pesci, Reggiani
Polo
Probability in Middle School
Teaching and learning problems in
Geometry. A preliminary study for the
construction of a didactical situation:
reference system and space geometry
Cognitive and metacognitive aspects of
problem solving in primary school
Introduction of basic ideas of computer
science in primary and middle school
Arzarello
Arzarello,
Bartolini,
Mariotti
Fasano
Boero
Bernardi,
Boero,
Gallo
Malara,
Marchini,
Morini
Malara,
Menghini,
Reggiani
P.Ferrari,
Reggiani, Zan
Bazzini,
Chiappini,
P.Ferrari
Title
Mariotti
Arzarello, Boero, Malara,
Mammana, Pellerey,
Prodi, Scimemi, Villani
Rocco Testa Trevisan
Arzarello, Bazzini,
Chiappini, Gallo,
Furinghetti, Menghini
Bartolini, Gallo, Speranza,
Grugnetti, Polo, Villani,
Zuccheri
P.Ferrari, Lemut, Mariotti,
Pesci
Accascina, Berneschi,
Bornoroni, De Vita, Della
Rocca, Olivieri, Parodi,
Rohr
Boero, Cannizzaro,
Crocini, Scali
Bottino, Chiappini,
Giannetti, Scarafiotti
Gherpelli, Iaderosa,
Malara, Navarra
Bartolini, Boero, Garuti,
Parenti
Arzarello, Mariotti,
Olivero, Paola, Robutti
Semantic fields in the teaching and
learning of mathematics: reflections on
the conceptualization and linguistic
mediation problems related to
curricular innovation experiences
The interplay between images and
concepts in geometrical reasoning
Reactors
Date
Margolinas,
Prodi
January February
1987
Barra, Pesarin July 1987
Bartolini Bussi, December
Villani
1987
Dapueto,
Janvier
Luccio,
Pellerey
May 1988
Arzarello,
Dapueto
May 1989
Ciliberto,
Pontecorvo,
Di Stefano,
Quattrocchi
January
1990
December
1988
Educational research and teaching
practice
December
1991
Teaching and learning of algebra: state Bernardini,
of things, methodologies of research
Toti Rigatelli
trendes and perspectives
Geometry: epistemology, research
Boero, Pellerey
methodologies, present trends
December
1992
December
1993
Representations in Mathematics and
teaching-learning processes
Aiello, Ferri,
Iaderosa
January
1995
Bridging problems in mathematics
between high school and university
P.Ferrari,
Marchini
October
1995
The learning of number in primary
school
Microworlds, hypertexts and
communication systems in mathematics
education
The problem of the transition
arithmetic-algebra in middle school
The first approach to theorems in
primary and middle school
Arzarello, Zan
Gazzaniga,
Mariotti
January
1996
December
1996
Chiappini,
Gallo
Arsac, Barra,
Polo
December
1997
December
1998
Introducing students to theoretical
thought: investigation on some
mediators
Bernardi,
Capponi, Pesci
December
1999
71
FRANCESCO SPERANZA AS A MATHEMATICS EDUCATOR:
VALUES AND CULTURAL CHOICES
Nicolina A. MALARA
Introduction
It is not easy to paint a picture of a scholar so recently after his death,
especially for one who considers herself, in a certain sense, one of his
students: there is a risk that one's affection might blur one's objectivity.
Therefore, in order to retrace the figure and the works of Francesco Speranza,
we shall make frequent reference to his very own words or, if this is not
possible, go back to the closest sources. Since it would be impossible to
express his thought completely in just a few pages, our aim shall be that of
drawing an outline, hoping that from a few brush strokes we are able to show
something of his great culture and humanity. Our description shall be
nevertheless "partial", in the sense that the things we highlight will inevitably
be those that we value most greatly and with which we identify most closely.
We shall concentrate on F. Speranza the mathematics educator with reference
to the works of C. Marchini in this book when looking at the figure of
Speranza the epistemologist (even though separating the different aspects of
his personality is over-simplistic and in many ways wholly inappropriate).
We shall start from his concepts and values on the teaching of mathematics
which are essential in order to understand many of his choices. Then we shall
discuss his text books, his teacher training books and his popularisation of
mathematics. We shall also consider some of his important articles and finally
we shall look at his works of widespread interest to the community of Italian
didactic researchers in the field of mathematics.
1. Concepts and Values
F. Speranza may undoubtedly be considered one of the forefathers of
mathematic didactics in Italy. His interest in this discipline started towards
the end of the 1960s. The arrival of structuralism, of which he was a strong
supporter (Speranza 1994), made him aware of the need for modernisation in
maths teaching. At the time, though very young, he was full professor at the
University of Messina where, apart from his course on differential geometry,
he also held one of the two courses on “Matematiche Complementari”
(Complimentary mathematics)1. His teaching was modern both in content2
and in method3 and kept the historical/critical aspects of teaching in view.
This position led him to concentrate more closely on the question of maths
teaching. At the time, the founding of the single middle school (1963) and
the incorporation of mathematics teaching with that of the other sciences
together with the plans to reform the secondary school (which came to a head
in the so-called "programmi di Frascati" (Frascati programmes) (De Finetti
1967)), brought the urgent matter of teacher training to the forefront.
In 1970, having moved to the University of Parma, he dedicated himself to
the problem through the local Mathesis4 group . It can be understood just how
important the matter was to him from the fact that despite his reserved nature,
he called regular meetings in which he would put himself at the centre of the
discussion group on the topic of the didactics of mathematics and set out the
first steps of in-class innovation. He saw the key element for change in the
teachers themselves. Convinced that it was necessary to change their
mentality, he pushed as hard as he could for the promotion of new ideas and,
more than anything, for the setting up of mixed working groups of school
teachers and university professors. To reform teaching methods, he promoted
a "wave" style model which would start from direct action on a handful of
teachers who, through direct influence on their colleagues, would spread their
knowledge throughout the school. The basic hypothesis was that with a
reasonable distribution of such teachers across the country, it would be
possible to regenerate the school from the inside5. In those years, perhaps for
the very revolutionary importance of so-called modern mathematics, he was
not the only one to believe in the need for debate between different sides with
the single aim of instigating real change6, not just in Italy. For example,
Varga (1976) wrote:
1
2
3
4
5
6
Such courses, introduced in the 1920s in Italy thanks partly to the works of F.
Enriques, were designed for the cultural preparation of teachers. They were
traditionally dedicated to core studies of a classical nature for pre-university teaching.
In those years, he tackled foundational and mathematical logic questions. Together
with his students, he even worked on the then-new categories theory, giving it
unifying value and reintroducing it into the currents of thought that had produced it.
Even at the beginning of his career days he would give out a variety of tasks to
students (reconstitution of a theorem, the search for counter-examples, the formulation
and demonstration of some simple results from a "poor" axiomatic system, the
construction of models of simple systems of axioms, checks to find on which axiom
certain properties depend). His lessons were an open forum for debate and were
extremely involving, even though they required a great amount of revision, reelaboration and refining of the work done in lessons.
Ancient Italian Society of Mathematics Teachers, founded in 1895.
This is the birth of the Italian model of the search for innovation. I remember a
"promotional" visit of his to the University of Modena to highlight his project which
was considered with scepticism by several mathematicians at the time.
Among the initiators of a bond between school and university, we must remember B.
De Finetti and L. Lombardo Radice (Ceccherini 1993, Menghini 1993)
67
"Content – what to teach – can be imposed on a teacher. It is less easy to
check up on the ways in which the content is organised within a curriculum,
and it is even less easy to check up on the way in which it is presented. That
which is least controllable is the most specifically pedagogic task of
organising the children's work. But this can be transmitted from person to
person, like something contagious."7
He was among the first to maintain that real innovation must start from
primary school and he was the first to set up on-going collaboration with
primary school teachers, involving them also in the universities. He was also
involved in the recently re-established middle school, taking part in the
working group to create new curricula (1979). His ideas are clearly present in
their framework which was based on the projectual ability of the teacher,
aimed at linguistic refining and reasoning, aimed at "mathematisation" but as
a stepping stone towards theoretical thought (just think of the "structural
relationships and analogies" theme), aimed at mathematical ideas and their
historical development (geometry, for example, is examined in three stages –
synthetic, Cartesian and transformational).
In one of his works of the time (Speranza 1980), he presents his ideas on
mathematical education during the pre-university years. This work, of which
we shall look at a significant extract, can be seen as a real manifesto of his
view of teaching mathematics. In this work he puts forward the following
points both clearly and convincingly:
• Everyone has the right to study mathematics.
• Mathematics must be taught sufficiently but seriously in every school.
• Mathematics must serve man.
• Mathematics is rooted in experience.
• Mathematics must develop autonomously.
• Mathematics must collaborate with other disciplines.
• The problems of other scholastic levels must be kept in mind.
• Official curricula must stimulate teachers' initiative.
At the time he was against the integration of mathematics and science,
however a decade later his position on the matter was more open and less
decided though he considered it a limitation to talk about such integration in
the name of a greater unification of thought (Speranza 1990a, 1992b).
He would always value the link between mathematics and other disciplines,
with constant reference to philosophy and linguistics which he considered as
rich in content as mathematics. He considered art similarly, stressing the
importance of esthetic training in mathematics too (1986a). Furthermore he
7
68
This idea of "contagiousness" with regard to teacher-pupil relationships is expressed
in a "romantic" style by Enriques (1921), noble figure of Italian mathematics and
culture and strong reference point for Speranza (1992a, 1994b).
would acknowledge the use of mathematical methods as a tool in other
disciplines, even those not strictly speaking scientific8.
He would reflect on these themes9, his fundamental values, in the work “La
scuola - una passione civile, un modo nuovo per costruire il sapere” (School –
a common passion, a new way to build knowledge), a title which expresses
well his social commitment (Speranza 1995a). In this work, he looks at the
"central role of the person" in the learning process (both when talking about
teaching teachers and teaching students) and underlines the understanding
acquired by researchers of the aspects of the psychology of learning.
On the manner, role & meaning of mathematics teaching
at different scholastic levels
In primary school it is not suitable to impose complex terminology, but it can be
just as harmful to deviate from the basic structures for fear of "being too
mathematical". During the middle school years these structures must be
thoroughly learnt – it would be wrong to remain on the level of concrete
exercises. In secondary school formal thought must be thoroughly learnt and
"abstract ideas" should be addressed.
Not giving children the right mathematics at the right age means depriving them
of their right (and this goes especially for those that come from the lower end of
the social scale – they are the least likely to make up later.) If teaching does not
keep these principles in consideration, if it trusts only in the pupils' ability to find
the "logical thread" between the notions handed out to them, it causes one of the
well-known disasters of our schools: for the majority of people, maths remains an
incomprehensible episode of their school life. Not only in traditional schools but
also in "easy" schools, which refuse outright to offer basic essential notions, are
selective of and harmful to student ability.
MATEMATHICS MUST SERVE MAN
It is a product of man, a tool created to satisfy his own needs (and sometimes also
for esthetic purposes). The children must understand (or rather, be led to
understand) what the aim is, and at least during their compulsory schooling,
mathematics teaching should be pleasant, stimulating. I have seen – as I think
you have all seen too – six or seven-year-old children leaving their games to
come and solve reasonably complex "logic" questions. They take it as a game,
but this makes for ever better learning. Children have no difficulties in working
up enthusiasm or getting involved. Often it is us adults that extinguish their
enthusiasm and thus distance them from intellectual involvement.
It has been observed that the spirit of research typical of scientists is the
natural perseverance of curiousity of children. I would also like to point out the
8
9
The humanistic-cultural value of mathematics is also stressed by L. Lombardo Radice,
another noteworthy educator (Ceccherini 1993).
It is interesting to note that Speranza considers these themes in articles for
commemorative collections for O. Montaldo (Speranza 1990) and G. Prodi (Speranza
1995), mathematicians who, like him, have put much effort into the renovation of
mathematics teaching in Italy.
69
importance of mathematics teaching to handicapped children, both because they
too have the right to an education (and the school should commit itself to this
firmly), and also because of the complexity of problems that they ask themselves.
MATHEMATICS IS ROOTED IN EXPERIENCE
Traditional teaching often tends to give mathematical concepts without adequate
motivations, it tends to present them as ready-made ideas. In this way, it is
difficult to achieve an important target of maths teaching: getting used to
abstraction, that is, organising outwardly different facts into unifying systems. It
is not that we need abstract ideas but we need to know how to perform the
procedures of abstraction.
In teaching it is necessary to draw mathematics from the experiences of the
pupils in a form suitable to their different ages. It would be right to embrace this
principle as "Mathematics and Reality", but for young children, a story can be
more motivating than a concrete problem (naturally, young children must too be
led gradually towards concrete problems). Reality and experience must be
understood as widely as possible in order to give value to mathematics itself.
(From "L'Educazione Matematica nell'Arco Preuniversitario" (Pre-university
mathematics teaching), in “L’Educazione Matematica” (Mathematics Teaching), suppl. 1, 16)
He traces how the very idea of mathematics teaching has changed with the
shift of attention from the teacher (as a bank of knowledge to be distributed)
to the students (as participants in the knowledge to be built up.) He reviews
the principle social changes and the procedures gone through in order to
"make school grow", while lamenting the unresponsiveness of the
institutions. He explores diverse issues usually considered opposite between
them. For example, he sustains the complementariness of the supposed
dichotomies "basic mathematics and applied mathematics", and "mathematics
for structures and mathematics for problems", stressing how the aspects of
practice (applied and for problems) are indispensable to give body to the
theoretical content and sense to their organisation. Again, on the dichotomies
"basic mathematics and advanced mathematics" or "mathematics and other
disciplines" he believes it essential for teachers at every scholastic level to be
aware of the ground covered at all other levels and also for them to have the
means to understand the basic ideas covered in other scholastic areas, not just
the scientific ones. He strongly supports the "humanistic" component of
mathematics, its interchange with areas of psychology, anthropology and
philosophy, and reiterates the importance of recognising the importance of
the cultural dimension that mathematics has in society.
It is interesting to note how in this article he deliberately does not talk about
mathematics teaching at a university level10. He discusses this theme in
10
70
At the foot of a certain point he poses the eloquent question "And as for university
mathematics?" which he leaves unanswered.
Speranza (1988a and 1988b) and briefly in Speranza (1989b), highlighting
that mathematicians must question themselves on the role of mathematics in
society and reflect on "who is interested in studying mathematics?" and
"which type of mathematics should be widely-known?" and "if and when our
theoretical knowledge is useful outside a strictly professional environment".
He maintains that a mathematical knowledge is important for many middle to
high level professional positions which give potential for autonomy and
creativity". Thus it is important to aim for a widespread presence of
mathematics in all types of schools with a strong cultural component
and that primary school teachers should be given a good grounding in general
maths and science. He concludes by calling on the university and the
mathematics degree course of which he laments the closure and the
technicalism writing (1998b):
…giving a slightly more cultural edge to the preparation of our graduates
could turn out to be of advantage to the presence of mathematics in our society
insofar as it would help them gain positions of greater responsibility and
above all because it would mark the beginning of a wider interest in
mathematics. Many people may wonder, and rightly so, how mathematics can
be given a more cultural edge. There are no simple answers: new proposals
must be put forward and tried out.
In the last decade of his life he did all he could to focus attention on the
importance of the Epistemology of Mathematics for teacher training. For
didactic research he set up seminars on epistemology (Speranza 1992c,
Ferrari & Speranza 1994) and published many works (see the following essay
by Marchini). He fought bitterly to create a space within the university for the
teaching of epistemology, but he ended up a solitary voice. The university
preferred to consider history among the raccomanded disciplines forthose
who intnded to go into teachingmathematics as a discipline characterised only
by its didactic nature, serving as an end in itself. In Speranza (1995b), he
stresses the need for a history of ideas, not of events, and laments the
unrecognised value of epistemology for those who want to enter the field of
mathematics.
Textbooks
Francesco Speranza is the only Italian scholar to have written textbooks for
every scholastic level, of which all are highly appraised though generally not
widely used. We shall describe them, differentiating them by scholastic level.
Textbooks for secondary schools
The most celebrated books are those dedicated to secondary schools, though
they are commonly considered textbooks for the élite (using them has become
71
an implicit declaration of high quality teaching). They consist of two
complete programmes for the five years of secondary school (from the age of
15 to 19) written in collaboration with Alba Rossi Dell'Acqua in the
following order:
• “Matematica” (Mathematics) (five volumes) published between '71–'74
with two additional volumes aimed at the upper years of the magistrali
(school for primary school teachers)(1973).
• “Il Linguaggio della Matematica” (The Language of Mathematics) (five
volumes) published between '79–'82.
The first of these projects, realised on the basis of the Frascati programmes, is
profoundly innovative in comparison with the other texts available at the
time. The layout is decidedly modern, the language of the text maintains a
structuralist approach throughout and the geometry shows an axiomatic
Hilbertian touch. There is great innovation also in the use of diagrams of
different kind, used also to communicate very subtle concepts such as
structural isomorphism and numeric broadening. Among the exercises, along
with application and chapter revision questions, there is a new type conceived
as encounters with future topics; a ground-breaking notion for the times. The
two volumes for the scuola magistrale also include an interesting novelty. In
the last part there are several chapters dedicated to the most advanced didactic
research at primary school level. He touches on the work of Piaget, on the
primordial topological structures which he treats synthetically. He also talks
about the Dienes studies and activities with logical blocks or multi-base
material as well as presenting the structure of the Nuffield project.
The second project differs from the first for several fundamental reasons.
First of all, there is a greater emphasis on linguistic and relational aspects of
mathematics, as suggested by the title. There is also a revised treatment of
geometry: the idea of a single axiomatic approach is done away with,
replacing it with an eclectic one which presents differing points of view11.
More space is given to geometrical transformations and the section on vector
spaces is redimensioned. However, the most interesting thing – and not only
from a methodological point of view – is that both synthetic and analytic
passages on the same topic are to be found side by side highlighting the
character of algebraic model of the latter.
Textbooks for middle schools
These are followed by the books for middle schools “La Matematica: parole,
cose, numeri, figure” (Mathematics: words, things, numbers, figures) in three
volumes (1984). The book, which is not only very interesting but also
esthetically pleasing, aims to paint a rich portrait of mathematics. The
language is refined, the exposition is detailed and the references are wide11
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The reasons for this change are discussed in several of his writings which we shall
discuss later on in detail.
ranging: from physics to biology and the sciences in general as well as from
the most diverse fields of study such as:
1 the history of art, with the use of photographs and paintings (Morandi (the
coordination of view points), Dalì (Platonic solids) Escher (topological
transformations), Durer, Duccio da Buoninsegna, Giotto, Masaccio,
Mantagna, Veronese, ... (the history of perspective);
2 the history of science, with chapters dedicated to the evolution in the
conception of astronomic theories as well as taking classic problems as the
basis for new branches of mathematics (i.e. for probability he looks at the
problems solved by Galileo, Pascal and for topology, at the problem of the
Koenisberg bridges etc.) and examining also the emerging disciplines (i.e.
statistics, information technology);
3 literature, (i.e. the mathematical study in the short story I Sette Messaggeri
(The Seven Messengers) by Dino Buzzati in the chapter on linear functions;
and extracts and variations for new viewpoints taken from Abbot's
Flatlandia and transferred to his own "Moebiuslandia" and
"Spherolandia").
It may be noted that great attention is paid to logic/linguistics teaching,
structural analogies and geometry, which is laid out to put its polyhedric
nature to the forefront. Unlike its predecessors, the book was not very well
received by teachers, most of whom did not have a mathematical background,
very probably because of its originality and complexity.
Textbooks for primary schools
For the initial stage of the primary school, (six to seven-year-old pupils) he
created a fascinating project together with Luisa Altieri Biagi called
“Oggetto, Parola, Numero” (Object, Word, Number) (1981), which has a
double structure of a book for teaching teachers and a book for teaching
students. In fact, the part aimed at students is of an extremely "hands-on"
nature, with around 250 taskcards to complete in the first year. The taskcards
are extremely stimulating from a visual, linguistic and conceptual point of
view. They concentrate on verbalisation and act as a basis for further
activities of reflection and written verbalisation (Further on we shall also
consider the teachers' book.).For the second stage, (eight to 11-year-old
pupils) he created the mathematics section of the scholastic subsidiary
“Imparare a Scuola” (Learning at School) (1990) together with C. Mazzoni
and P. Vighi.
Methodologically, the two works revolve around grasping relationships in
different contexts, codifying them using different representations and
reflecting on different situations to trace analogies between them and pick out
underlying patterns. Constant attention is paid to natural language, to its
refinement and to the construction of mathematical language. However, the
difference between the two is enormous: the collection of taskcards is both
73
culturally refined and esthetically pleasing. The subsidiary, a typical
“ministerial” product, devotes a limited number of pages to the subject and
his section is generally poor compared to the taskcards.
Books for teachers and for a wider public
As we have seen, the problem of teacher training and more widely of
spreading a more accurate image of mathematics across society is very close
to our author's heart. There are several books specifically aimed at teacher
training. The first (Altieri Biagi et al. 1979), is a book aimed at middle school
teachers which attempts to highlight the links between different subjects. It
contains contributions from different authors and Speranza's article concerns
the theme of crossover between mathematics and language.
The following book is very important. “Oggetto, Parola, Numero” (Object,
Word, Number) (1981), written together with M.L. Altieri Biagi is aimed, as
mentioned above, at teachers of the first stage of primary school. The book is
split in two parts. The first part, devoted to interdisciplinary elements and
activities, gives a didactic itinerary starting from the observation of nature in
which objects (stones, leaves, ...) are studied and classified according to
different characteristics. There is then also a section elaborating on the
observations and reflections of what has been discovered. The itinerary is
accompanied by a teacher's commentary and the presentation of the relative
taskcards. The second part is devoted to aspects and activities related to the
environment ("object"), to language teaching ("word") and maths teaching
("number"). As said in the preface:
... the principle difference (between parts I and II) lies in the fact that in the
first part those topics (of nature observation, language, mathematics, history
etc.) were necessary in order to acquire "logical abilities". In the second part,
the "logical abilities" are needed to acquire certain skills for the discipline.
The treatment in the text operates on various levels: the theoretic content, the
cultural premises, the implications for pedagogy as well as the reflections on
the cultural project underlying the contents as they are treated; the
frameworking of the activities from a logical-structural point of view; and the
advice to teachers on the importance of knowing and keeping in mind the
goals set, and the general teaching scheme. Apart from the subjects discussed,
and they are discussed flawlessly, the book makes for easy, involving reading
with elegant diagrams and illustrations, fascinating for the plurality of
approaches and levels with which the same topic is treated.
Other important texts of note are “Matematica per gli Insegnanti di
Matematica” (Mathematics for Maths Teachers) (1983) and “Insegnare la
Matematica nelle Scuole Elementari” (Teaching Mathematics in Primary
74
Schools) (1986), the latter written together with D. Medici Caffarra and P.
Quattrocchi. These texts are conceived to offer a grounding in mathematics to
teachers without any specific training in the subject. The former is
particularly aimed at the preparation for the public exam to become middle
school teachers. The topics presented range from "classic" ones (arithmetic,
algebra, analysis, geometry) to other more recent ones (set theory and logic,
probability, statistics, elements of numeric analysis). The layout, however is
not that of a manual. The text starts and ends with chapters on the history of
mathematics to give an impression of the subject in its evolution, the
introduction to the different topics is always put in context and motivated,
and furthermore there are interesting problems to be solved.
The second, published to coincide with the new primary school curriculum,
aims first of all to give an idea of what mathematics is and what it deals with.
The first chapters are devoted to the crossover between language and thought,
the meaning of mathematical knowledge and the role of demonstration
compared to experimental science, and the question of confutability. There
follow several extracts taken from these books which help us understand our
author's social commitment and also his disdain for the direction that
schooling had then taken.
If the child is not stimulated at the right point to observe, to recognise, to
manipulate objects both concretely and conceptually, to make connections, to
generalise and make abstract, his or her logical potential may be blocked and
may not easily be unblocked. If the child is not used to interacting with
reality, (with the world of things and of man) to modifying it, he will then be
unlikely to develop projectual capabilities.
The responsibility of adults (the family, the school, society) is therefore very
great in this delicate period. And it is incredible that in a modern society, as
we boast ours to be, does not dedicate all of its energy, resources and
economic power to the training of its educators, for example, to the
professional training of those involved in this sector.
(From the preface of “Oggetto, Parola, Numero”, (Object, Word, Number))
Mathematics today is the great unknown. Despite being present in nearly all
school curricula, only a few teachers have had the chance to form an accurate
idea of what it really is today. Most maths teachers in compulsory education
have not had even the bare minimum of necessary training....
This sad situation is intolerable for our society, given the importance that the
years of childhood and adolescence have for the development of a person,
and for the importance that mathematics has both from an educative point of
view as well as from that as a language and instrument of knowledge.
(From the preface of “Matematica per gli Insegnanti di Matematica”
(Mathematics for Mathematics Teachers))
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Works for a wider public
As far as popularisation is concerned, we may note:
• His first text “Relazioni e Strutture” (Relationships and structures) (1970).
A welcoming invitation to take up mathematics, it was conceived with the
aim of spreading a knowledge of the fundamental topic areas of so-called
"modern mathematics" (from simple set theory, relationships and
properties, graph theory, algebraic and topological structures to the most
general concepts of mathematical structures and the isomorphism of
structures).
• The treatment in the ample entries "Cos'è la Matematica" (What
mathematics is), "Logica" (Logic), "Geometria" (Geometry) and "Le
strutture Matematiche" (Mathematical Structures), from the “Enciclopedia
delle Scienze” (The Encyclopedia of Science), De Augostini (1984), and
the supervision of the entire mathematics section, and lastly an entry in the
“Enciclopedia della Scienza e della Tecnologia” (The Encyclopedia of
Science and Technology) (Speranza 1994c).
There are also countless articles written for teaching journals for all levels of
schooling (in line with his values and beliefs), which there is no space here to
discuss. For these, we might suggest turning to the bibliography of his works
compiled by himself for the convention held on his 65th birthday (D'Amore e
Pellegrino, 1997).
A look at some of the articles
Along with the above mentioned articles, we must also consider the articles
written at academic levels and published in journals or in proceedings .
Because of the sheer number of them, we shall choose to cite just a few of
them, restricting the choice to just two fundamental themes in his career:
Linguistics (Speranza 1980, 1982, 1986, 1989, 1994, 1996) and Geometry
(Speranza 1986, 1988, 1989, 1992, 1994, 1995, 1997)12 which seem to us to
best express his opinions. We shall look at several of these: to be precise, one
on linguistics and four on geometry.
Works on linguistics
Out of the various articles referred to above (and those are not all of them),
we have decided to look at “Dal linguaggio naturale al linguaggio
formalizzato: le variabili” (From natural language to formalised language:
the variables) (1982) below for the importance given to the concept of the
12
76
In this choice we have referred also to important journals, widely available. However,
there is not always a clear distinction between Linguistics and Geometry, for example,
Speranza (1996) can also be classed under Geometry and Speranza (1986) under
Linguistics.
mathematical variable and that in his didactics. In this article, also interesting
from a historical point of view, he considers variables in diverse contexts:
• variables in language (he offers an interesting and sharp analysis of an
extract from I Promessi Sposi (The Betrothed) in which he looks at a wide
variety of pronouns) and the constant/variable slide (using a humourous
extract by J.K.Jerome),
• variables in geometry (he compares two extracts, one taken from Plato's
Meno and one from Euclid's Elements to underline the shift from a
discursive to a formal level with the introduction of names for
indeterminate objects, points, lines, etc. which, by the very nature of their
arbitrariness come to be considered variables),
• variables in logic and set theory (he compares two extracts, one taken from
“Analitici Primi” by Aristotle and the other taken from the “Summulae
Logicales” by Pietro Ispano which both examine a certain type of
syllogism, (Baroco), which entails reduction to absurd levels in order to
show the greater value of the Aristotelian formulation compared to that of
the scholastic tradition. That is, by overlooking the reference to meaning, it
generally allows one to work safely and systematically.
• numeric variables (4,000 years ago, 1,500 years ago, 400 years ago: he
examines the ways of expressing numbers which satisfy the conditions
given in the Babylonian era and by Diofantus. Extracts from Tartaglia,
Bombelli, Vieté, Harriot and Descartes concerning the resolution of third
degree equations in order to observe the greatest synthesis and
expressiveness in representation with the introduction of symbolic writing).
With this outlook, he gives a tangible impression of the pervasiveness and
evolution of the concept. In the conclusion, he stresses that natural language
and mathematical thought are instruments that respond to the need to
represent experience, to communicate, to think and act upon the world. He
voices the possibility, from a didactic point of view, to use an early, selective
introduction to letters to allow for a more conscientious approach to algebraic
symbols.
Works on geometry
With reference to geometry, on which he wrote a great deal, we shall restrict
ourselves from the selection given above to a few far-reaching articles which
show to the full the complexity and moderness of his thought and which are
also of importance for didactic research.
The first, “La Razionalizzazione della Geometria” (The Rationalisation of
Geometry) takes the case of geometry teaching and declares it to be rapidly
worsening13. He proposes a reappraisal of the subject right from primary
13 He attributes this drop in standards to the type of university teaching too for its
sacrifice of an all-round view in the name of formal elegance.
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school, according to the spirit of the curricula aimed at building geometric
knowledge based on experience and "know-how". He offers some cultural
and didactic advice on how to build geometry as a rational science, placing
the accent "...on the idea of constructing the organisation of geometry, in
contrast with the tendency to give it as 'something already sorted out'.". He
recalls the changes of conception of the origins of geometric knowledge,
underlining that:
• many 19th and 20th century mathematicians/philosophers (Gauss,
Riemann, Mach, Enriques) saw such origins in experience, but by giving
value to the human mind which constructs theories by interacting with
experience;
• Piaget took a similar line with regard to the development of intelligence;
• compulsory schooling curricula are based on this concept;
• even in secondary school it would be right to maintain a certain contact
with experience, both in the approach to theory and also to highlight how
this can allow us to interpret experience better;
• it is important to study axioms for their logical/organisational value to
knowledge;
• it is important to offer critical and historically-founded teaching and show
the cultural extent of Erlangen's programme.
He highlights the role of definitions and shows how their use is not at all
banal giving an analysis of eventual redundancies. He then focuses the
attention on affirmations, writing:
... this area must also be initially approached informally; this is true for
compulsory schooling, but also to some degree for secondary schools.
Especially in the latter, too much importance is given to the moment of truth,
(the demonstration) compared to the moment of discovery (this is what we are
taught to do in Euclid's' text which starts off with the terms of a problem
without explaining how we got there, and then moves on to the
demonstration). Instead, at all levels of schooling students must be pushed at
least every so often to make some suppositions – if some of them are
incorrect, all the better (after all, even mathematics usually progresses through
suppositions which are only later proved wrong or demonstrated or remain
suppositions. One must therefore work on geometry as if it were an
experimental science: the suppositions must undergo "experimental checks",
generalisations must be tried out. Obviously, these "checks" cannot give any
certainties: in this way one understands better the need for demonstrations.
This last point could be considered the conceptual basis of much of the
current Italian research and not only on geometry (see the essay on the
national research seminar and in particular the papers relative to the seminars
of the last few years).
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In the second article, eloquently entitled "Salviamo la Geometria" (Save Our
Geometry), Speranza picks up the previous themes but from a more didactic
point of view, focussing on how geometry suffers from a multiplicity of
traditions:
• the artisan-artist tradition, of geometric "know-how", (from the
construction of pyramids to that of medieval cathedrals), or of pictorial
representation (which, by giving birth to the theory of perspective, open up
new directions);
• the Egyptian/Babylonian tradition, probably borne from land surveying
which naturally leads to the calculation of areas and volumes);
• the critical tradition of constituted knowledge, (the school of Pythagoras
and the discovery of the unmeasurable, Zenone and paradoxes, Galileo and
the infinite, the history of criticism, the postulates of parallels and the
consequential revolution in the concept of geometry);
• the Euclidean tradition, which values the organisation of knowledge (which
is supposedly definitive) and the recent frameworks (as children of the
critical tradition) of Peano, Pieri and Hilbert;
• the Cartesian tradition, which leads to an interpretation of geometry in
terms of algebra (and also to give algebra a sprinkling of geometry);
• a physics/astronomics tradition, which draws on the applications for natural
sciences and sees the construction of mathematical models as being of
geometric nature (Eudoxus, Apollonius-Tolomeus, Aristarcus-Copernicus
up until the relativist cosmologies).
He maintains that these traditions can and indeed must go hand in hand with
geometry teaching through a curriculum designed across the pre-university
years right from primary school. In actual fact, they are often lumped together
without critical comment leading to didactic distortion and educational
damage.
He then gives a view of the development of geometry in correlation with the
development of intelligence, showing which of the different traditions
intervene and how to refer to them in teaching. He sums up everything in an
interesting table from which the links between different didactic activities at
different stages of thought development are shown.
The third article considered here is "Controindicazioni al Riduzionismo"
(Side Effects of Reductionism). It is a complex work in which he dwells both
on the problem of degeneration from a didactic/cultural point of view and
also on that of the reduction to the algebraic model in the current treatment of
geometry. He does not declare himself to be against reductionism understood
as the co-ordination of different points of view with which to examine the
same theory as if it were an appendix of one of its own models. He would
always analyse from an epistemological point of view the inter-weaving of
paths which, right from the introduction of the co-ordinates method, has
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brought the current work on algebra affine (and projective) spaces in one
field. He then underlines the limits of this reduction which, incidentally, does
not allow one to master affine geometry as a whole (for example it cannot
cover non-Desarguesian planes). He concludes reiterating as in his other work
that the only possible solution is not to give in to temptation to follow a nice,
single presentation but, on the contrary, bare different points of view in mind
in order to let students grasp geometry in its complexity.
The fourth article, "Alcuni nodi concettuali a proposito dello spazio" (Some
conceptual hitches about space), which formed his contribution to the ninth
session of the national seminar, examines concepts of space understood also
in terms of the implicit philosophies with which it is looked at. He starts off
by invoking Erlangen's programme which orders the different types of
geometry in relation to the basic groups of transformations with which they
are associated. He maintains that each geometry is connected to a significant
field of experiences (even mental ones) which leads one to operate according
to the corresponding group and in relation to which one may develop a
certain intuition. He states that he considers the distinction between space and
plane to be artificial. He declares:
…However, I shall not counter 'space' against 'plane'. The environment in
which we operate might even be two-dimensional. I must clarify a point here,
many people say that physical reality is at any rate three-dimensional so the
three-dimensions must be given 'right of way". I do not agree with this
conclusion – even when we act physically, we operate however on the level of
mental models and these might be two-dimensional. Dimensions are therefore
a feature of mental images of space.
On the concept of space he states
When we talk about concepts of space, we don't mean mathematically
elaborate rules of theory, in fact, sometimes these are not even clearly
expressed. But it is for this very reason that these "implicit philosophies" can
heavily condition our way of thinking (and condition the communication
between teacher and pupil). They might date back to a "primitive" thought (in
a psychological or ethnological sense), or even to pre or extra-curricular
teaching/learning. The teacher (and the researcher) must be aware of his/her
own concepts and analyse them (and basically be open to everything), while
being prepared to come across different concepts from the students, and
different indeed from student to student.
He develops the theme by opposites, comparing these themes and the story of
their evolution:
1) non-independent space, independent space;
2) absolute space, relative space;
80
3) homogenous or non-homogeneous space, - isotrope or anisotrope space;
4) limited, unlimited space;
5) finite or infinite space; space as a collection of points or as an irreducible
continuum;
6) real space or the projection of our senses (or…);
by looking again at each of these in the Erlangen programme.
He underlines the sharp contrast between the components of the last pair of
concepts and how the choice of one of the two influences and, at times, even
determines the choice of a component in relation to pairs considered
previously. He poses the issue of a "third way" between the two concepts, in
order to pass over Kant's vision with the birth of non Euclidean geometry and
declares himself, like Enriques, a supporter of "experimental rationalism"
which is also important for the teaching/learning process insofar as:
…space is not something ready and waiting which needs to be learnt and
neither is it a pure and simple mental (or neurological) structure; it must be
formed using suitable strategies.
This thesis is bound up in a long and well-documented treatise on the
concepts of space in art. He analyses this shift from non-independent to
independent space (which occurs in Tuscan painting and in the mathematical
theorisation of perspective) using the studies of Panowsky, Fracastel and
others, and underlines how certain themes (the idea and the representation of
the infinite, aspects of analytic geometry, transformations etc.) have always
been treated much earlier in art than in mathematics.
Other important works which complete the description of his writings on the
crossover between geometry, epistemology and didactics, which for reasons
of space we shall not look at here, are those dedicated to the cultural
significance of the Erlangen programme (1992) and of non-Euclidean
geometry (1997).
Collaboration and collective works with regard to "Seminario
Nazionale".
He took part in several of the didactic research groups' initiatives. In
particular, we may remember his participation in the book-writing group to
document Italian research projects into mathematics teaching at the following
congresses: PME 15 (Barra et al. 1991), ICME 7 (Barra et al. 1992), ICME 8
(Malara et al. 1996). In particular, we must not forget the contribution given
to the organisation of the task of reconstructing the history of didactics in
Italy from unification to the present day which forms the first part of Barra et
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al. (1992). His kindness, modesty and the care with which he set about his
various different commitments without ever throwing his weight around must
also be remembered. We might also remember his active participation in the
Working Group Mathematics Education as a Scientific Discipline at the
ICME 8 Congress (Seville 1996) at which he made a great contribution with
the presentation of an essay on the role of epistemology in mathematical
didactic research (Speranza 1997c), his synthesis of the records of work done
(Malara et al. 1997), and a further essay based on the vision held in the group
by E.C.Whitmann, Didactics of Mathematics as Science Design. In this essay,
Speranza gives an indication of how to construct an epistemology of Science
Design on the basis of significant ideas from the Science of Knowledge, with
interesting implications for didactics (Speranza 1997d).
And finally,
We are aware of the inadequacy of this synthesis compared to the vast and
worthy output of our author. We feel it is right to say that the Italian public
owe a lot to him, especially for the devoted work done by him to enlarge the
cultural dimension of mathematics, and for the attention paid to
philosophical, human and social elements of the discipline. In the article
following this by C. Marchini, though somewhat fragmented for the number
of themes touched upon, the figure of a man of great culture and human
sensitivity apart from the mathematician shines through. For this reason, as
we conclude we would like to dedicate to him, with great affection and
consideration for that which he has left us, a passage which would seem to
suit him faithfully:
A man sets himself the task of designing the world.
As the years go by, he populates a space with images of provinces, kingdoms,
mountains, bays, pools, islands, fish, houses, instruments, stars, horses and
people.
Not long before dying, he discovers that the patient maze of lines traces the
image of his own face.
(From "Epilogue” in Artifice by J.L. Borges, 1996)
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References
Papers by F. Speranza quoted in the text
ALTIERI BIAGI, M. L., PASQUINI, E., SPERANZA, F.: (eds) Per una didattica
interdisciplinare nella scuola media, Il Mulino, 1979
ALTIERI BIAGI, M. L., SPERANZA, F.: 1981, Oggetto, Parola, Numero, Itinerario
didattico per gli insegnanti del primo ciclo, N. Milano , Bologna
ALTIERI BIAGI, M. L., SPERANZA, F 1981, Oggetto, Parola, Numero, schede di
lavoro, N. Milano , Bologna
FERRARI, M., SPERANZA, F (eds): 1994, Epistemologia della Matematica,
seminari 1992-1993, project TID-CNR FMI series, vol. 14
MAZZONI, C., SPERANZA, F., VIGHI, P.: 1990, fasci colo iniziale, 3, 4, 5, in
Frabboni, F, Speranza, F (eds), Imparare a Scuola, N. Milano, Bologna
SPERANZA, F.: 1970, Relazioni e Strutture, Zanichelli
SPERANZA, F.: 1980, L’educazione matematica nell’arco pre-universitario, in atti
Convegno “Nuovi curricoli e metodologie dell’educazione matematica
nell’arco degli studi universitari” (Cagliari) L’ Educazione Matematica,
maggio 1980, suppl. 1, 1-6.
SPERANZA, F.: 1983 , Matematica per gli insegnanti di Matematica, Zanichelli,
Bologna
SPERANZA, F.: 1984, 1. Che cos’ è la Matematica; 2. Geometria, 3. Logica, 4.
Strutture Matematiche, in Enciclopedia delle Scienze, De Agostini, Milano, 522; 61-87; 22-37; 190-196.
SPERANZA, F. , MEDICI CAFFARRA, D. QUATTROCCHI, P.:1986, Insegnare la
Matematica nella Scuola Elementare, Zanichelli, Bologna
SPERANZA, F.: 1980, La linguistica e la matematica, Educazione Matematica, vol.
1-2, 46-54
SPERANZA, F.: 1982, Dal linguaggio naturale al linguaggio formalizzato: le
variabili, Educazione Matematica, suppl. III, n. 1, 123-138
SPERANZA, F., 1986, Le radici comuni di lingua e matematica in Altieri Biagi
M.L. (ed) Insegnare Lingua Italiana con i nuovi programmi nella scuola
elementare, Fabbri.
SPERANZA, F., MEDICI, D., VIGHI, P.: 1986b, Sobre la formaciòn de los
conceptos geométricos y sobre el léxico geométrico, Enseñanza de las
Ciencias, vol. 4, n;1, 16-22
SPERANZA, F.: 1988a, Osservazioni sul riordinamento del corso di laurea in
Matematica, La Matematica e la sua Didattica, anno II, 62-64
SPERANZA, F.: 1988b, Quale Matematica, La Matematica e la sua Didattica, anno
II, n.2, 63-64
SPERANZA, F.: 1988c, Salviamo la geometria!, La Matematica e la sua Didattica,
anno II, n. 2, 6-13
SPERANZA, F.: 1989a, Matematica e Linguaggio, Educazione Matematica, Anno
X, serie II, vol. 4, n. 2, 97 – 114
SPERANZA, F.: 1989b, La razionalizzazione della Geometria, Periodico di
Matematica, VI, 65, 29-46
SPERANZA, F.: 1990a, Matematica e Scienze, quale distinzione, quale integrazione
L’Educazione Matematica, anno XI, serie III, vol. 1, suppl n. 2, 47-54
SPERANZA, F.: 1990b, Controindicazioni al riduzionismo, La Matematica e la sua
Didattica, anno III, 12-17
SPERANZA, F.: 1992a, Il progetto culturale di Federigo Enriques, in D’AMORE, B,
PELLEGRINO, C. (eds), atti Convegno per i sessanta anni di Francesco
Speranza, univ.Bologna, Bologna,
SPERANZA, F.: 1992b, Tendenze empiriste nella Matematica, in Speranza, F. (ed),
Epistemologia della Matematica, seminari 1989-1991, project TID-CNR FMI
series, vol. 10
83
SPERANZA, F.: 1992c (ed), Epistemologia della Matematica, seminari 1989-1991,
project TID-CNR FMI series, vol. 10
SPERANZA, F.: 1994a, Attualità del pensiero di Federigo Enriques, La Matematica
e la sua Didattica, vol. 8, n.2, 112-132
SPERANZA, F.: 1994b, Linguaggio e Simbolismo in matematica, in Iannamorelli,
B. (ed) Insegnamento e apprendimento della matematica: linguaggio naturale
e linguaggio della Scienza, Atti I seminario Internazionale di Didattica della
Matematica, Sulmona, 1993
SPERANZA, F.: 1994c, La Geometria: da Scienza delle figure a Scienza dello
Spazio, in Enciclopedia della Scienza e della Tecnologia, De Agostini
Milano, 551-553
SPERANZA, F.: 1994d, Alcuni nodi concettuali a proposito dello spazio,
L’Educazione Matematica, anno XV, serie IV, vol. 1, n. 2, 95-115
SPERANZA, F.: 1995, La “rivoluzione” di Felix Klein, L’Insegnamento della
Matematica e delle Scienze Integrate, vol. 18B, n.4, 328-345
SPERANZA, F.: 1996, Il Triangolo qualunque è un qualunque triangolo?
Educazione Matematica, anno XVII, serie V, vol. 1, n. 2-3, 13-28
SPERANZA, F.: 1992, Il progetto culturale di Federigo Enriques, in D’Amore, B.,
Pellegrino, C. (eds) Convegno per i sessanta anni di Francesco Speranza,
Bologna
SPERANZA, F.: 1995, La scuola: una passione civile, un modo nuovo per costruire
il sapere, L’Insegnamento della Matematica e delle Scienze Integrate, vol.
18A-19B, n.5, 520-531
SPERANZA, F.: 1997, Scritti di Epistemologia della Matematica, Pitagora editrice,
Bologna
ROSSI DELL’ACQUA, A., SPERANZA, F,: 1970-1974, Matematica per le Scuole
secondarie Superiori, voll. 1-5, Zanichelli, Bologna
ROSSI DELL’ACQUA, A., SPERANZA, F,: 1973, Complementi di Matematica per
gli Istituti Magistrali, voll.1 - 2, Zanichelli, Bologna
ROSSI DELL’ACQUA, A., SPERANZA, F,: 1979-1982, Il linguaggio della
Matematica, voll. 1-5, Zanichelli
Other References
BARRA M., FERRARI M., FASANO M., FURINGHETTI F., MALARA N.A.,
SPERANZA F. (eds): 1991, Some Italian Contribution on Psychology of
Mathematics Education, CSU, Genova
BARRA M., FERRARI M., FURINGHETTI F., MALARA N.A., SPERANZA F. (eds):
1992, Italian Research in Mathematics Education: Common Roots and Present
Trend, TID- CNR project, FMI Series, vol. 12, 9-51CECCHERINI, P.V., 1993,
Lucio Lombardo Radice: una commemorazione, Lettera Pristem-Dossier, n. 2, I-VII
DE FINETTI, B.: 1967, Le proposte della Matematica per i nuovi licei, Periodico di
Matematiche, serie IV, XLV, 75-153
D’AMORE B. E PELLEGRINO, C. (eds), 1997, Atti del Convegno per i
sessantacinque anni di Francesco Speranza, Pitagora Editrice, Bologna
ENRIQUES, F.: 1921, Insegnamento dinamico, Periodico di Matematiche, serie IV,
vol.1, 7-16
MENGHINI, M., 1993, Lucio Lombardo Radice, Cultura e Metodo, , Lettera
Pristem-Dossier, n.2, VII-IX
MALARA N.A. MENGHINI M, REGGIANI M. (a cura di): 1996, Italian Research in
Mathematics Education: 1988-1995, Litoflash, Roma
Malara N.A. (a cura di): 1997, An International View on Didactics of Mathematics
as a Scientific Discipline, AGUM, Modena
VARGA, T.: 1976, La riforma dell’insegnamento della matematica,
L’Insegnamento della Matematica e delle Scienze Integrate, vol. 11, n.7/8, 705-714
84
THE PHILOSOPHY OF MATHEMATICS
ACCORDING TO FRANCESCO SPERANZA
Carlo MARCHINI
Introduction
F. Speranza dedicated the last years of his life to the analysis of the
knowledge of mathematics and its development as a discipline. This analysis
can be defined as epistemological and with it he succeeded in (re)introducing
these subjects into the Italian Mathematical Community, thus (re)establishing
a new interest and new forms of co-operation with a more strictly
philosophical environment.1
Starting from his mainly bourbakist style, which shows in his scientific works
and school texts from 1971 on, I tried to grasp some aspects in the evolution
of his thought by analyzing his published works and some manuscripts, with
the intention of offering a personal interpretation and encouraging further
studies on these subjects. Scritti di Epistemologia della Matematica Speranza
(1997b) (Papers on epistemology of mathematics) was to me of great help: it
is a collection of 18 essays in chronological order from 1987 to 1995 and it
contains a bibliography (incomplete) of his works on epistemology of
mathematics starting from 1986 and ending with some works appeared in
1998, the year of his death. Speranza chose the texts of this collection
himself.
2. The philosophy of mathematics.
In all of Speranza's production on epistemology of mathematics we find a
strong interest in mathematics education. Didactics has a two-sided role: it is
the 'source' of the problems Speranza wishes to analyze from a different point
of view, with a philosophical perspective, and it is the field of application of
the examples he mentions in almost all his essays. Moreover, Speranza gets
several ideas from experimental studies carried out in class. The development
of his intellectual position is not self-centred, on the contrary it is spurred by a
'social' need to provide ideas, tools and methods for teachers.
In Speranza (1987) he writes:
1
The Epistemology of Mathematics was widely developed by Peano and his school,
Enriques, Vailati, etc. and in Parma by Pieri and Beppo Levi. Political and cultural
events in Italy caused an interruption in these studies which were started again only
recently.
"I believe my personal experience might be of some interest. I have been
studying mathematics education for nearly twenty years, first for High School,
then for Junior High School and then for Elementary School. When I reached
that stage I realized that mathematics, in its ordinary perspective, did not
provide the needed support in some fundamental decisions to be made; it
called for some epistemological, I shall say philosophical with no intention to
scare, type of choice."
And in Speranza (1990) he states:
"When you draft a didactic plan you immediately come across important
choices to be made: for example what approach you should have to geometry
in Elementary or Junior High School; what type of logic is to prevail, what
type of interaction with other fields of learning is to be developed; ... You
realize that the results inherent to mathematics are not enough to make a
choice, that you need a wider, epistemological view ... .
In other words, the philosophy of mathematics is necessary to devise and draft
a didactic programme in compulsory education ... In High School the
technical aspects of maths can be more significant, however we know that the
new curricula have stirred up some discussions which can lead us to important
epistemological concepts: the interaction between logic and other sectors, the
construction of geometry, the role of non-Euclidean geometry, the meaning of
probability.
In fact, at this stage it is desirable for epistemological thinking to become
explicit since the first two years of High School ..."
And in Speranza (1988) he states:
"I believe that all these traditions can and should play an important role in
programming Geometry teaching. However, it seems to me that unfortunately
they are too often used (and combined) in a dogmatic way: instead of
releasing positive effects, they can cause didactic distortions.
This may happen both by making senseless combinations without a guide line
and by insisting on only one of them"
Some of his 'early' works on epistemology are meant to 'legitimize' his interest
in philosophy, by locating his position as a continuation of the tradition of
mathematicians who dedicated themselves to philosophy starting in the 19th
century and especially of the re-elaboration of the ideas of Federigo Enriques,
famous mathematician brought up as example for his doings in the field of
philosophy. Speranza shows interest in Federigo Enriques's ideas in all his
works. One of his last works was the editing (with O. Pompeo Faracovi) of
the acts of a convention held on Enriques in Livorno in 1996, printed in 1998.
84
Speranza considers Enriques a forerunner of several ideas that were
developed later on. To Enriques he ascribes the anticipation of historic
epistemology (Lakatos) and genetic epistemology developed by Piaget. In his
following works it seems he has overcome the problem of presenting
epistemology, or rather, satisfied with the success of the meetings he was
holding, he appears to develop his own autonomous position, which is clearly
recognisable even within the field of philosophy. In Speranza (1990) once
again he outlines, with a more confident tone (I think), the role played by
mathematicians and physics in epistemology from Kant onward and argues
that present field distinctions are historic accidents to be overcome. This
would answer a basic source of dissatisfaction:
"Contrary to the majority of scientific ambits, the idea that science is (or
should be) something useful has nothing to do with mathematics... The answer
we do maths pour l'honneur de l'esprit humain, which is basically right,
generates a form of dissatisfaction if there is no reference guide line"
The discontent is connected to the fact that as mathematical studies increase
and spread:
"...the number of people capable of understanding a certain work decreases..."
From a didactic point of view in Speranza (1988) he states:
"It might seem a paradox, but it is not: if you increase the educational level
too much, the product - that is what the students actually assimilate decreases, in fact beyond a certain extent it literally collapses".
Even the doubts on the role of mathematicians seem to be a reason for
discontent, as it is outlined in Speranza (1989):
"We mathematicians should ask ourselves if and when our theoretical
knowledge is useful to us outside the mere professional sphere"
The philosophy of mathematics can eliminate this dissatisfaction. In Speranza
(1990) he can single out three spheres: genetic epistemology, for investigating
the origins of mathematical knowledge, by exploiting both the support of
psychology and of the more strictly philosophical (ontological) nature of
mathematical entities in connection with the problem of the universals.
History, especially rational reconstruction of history as supported by Lakatos.
The third ambit concerns foundation problems.
"These ambits can interact with one another. A certain trend, from Frege to
Popper, considered psychology as unrelated to epistemological problems (thus
leaving the genetic sphere aside).
85
I believe we must be really careful with this type of exclusive selection, which
can have a personal value at the most. An interaction between the various
levels seems to be highly desirable...".
In my opinion Speranza's contribution to the field of philosophy is that of
having brought up the problem of the empiricist component of mathematics 2.
He fosters his position in Speranza (1992a) and proves it with a long series of
examples taken from the history of mathematics up to Enriques' experimental
rationalism and Lakatos' later re-elaboration of this issue. He then analyzes
two subjects, which appear again more precise in later works, that of
revolutions and the falsification in mathematics. The idea of revolutions
appears in a famous work by Kuhn (1962) where there is no mention of
mathematics; on the contrary, mathematics is said to have had paradigms
since ancient times. The issue of revolutions is then developed in three works:
Speranza (1994b), Speranza (1994a) and Speranza (1994c). In 1992 at the
same time three works appeared, Speranza's mentioned work, an article by
Ernest (1992) and a book by Gillies (1992) where the issue of possible
revolutions in mathematics is discussed. Speranza, however had long before
come to think that it was possible to locate revolutions in mathematics. In the
already mentioned Speranza (1987), appeared in 1987, he states:
"It is momentous that the Erlangen programme dates back to 1872, a decisive
moment for another essential step of the geometric (or rather metageometric)
thought: the proof that non-Euclidean geometries are coherent. I believe this
to have a revolutionary value as significant as the other mentioned result".
The idea of revolution itself is meaningless if set outside a historic
epistemology and a non-absolute philosophy. There are very different
positions on this matter. The objection set forth to the presence of revolutions,
in Kuhn's thought, is presented, though still not completely assimilated in
Speranza (1994a):
"According to the traditional view, mathematics proceeds by accumulating
results: is not Greek mathematics still considered valid ?"
This is the position of the so-called continuists (among them Enriques).
Speranza, in the ranks of the revolutionaries, objects that at present the
validity of Greek mathematics has deeply changed, moving away from the
guaranteed true science or knowledge
What is stated in Speranza (1994a) is highly significant:
2
86
The term empiricism in the present philosophical scene needs further specification.
Even 1ogic positivism is referred to as neo-empiricism, however I do not think
Speranza accepts this definition; in fact in a manuscript, Speranza (1998b), he moves
away from this school of thought. The manuscript is an incomplete draft.
"I believe that the idea of revolution in mathematics can be a useful tool for a
philosophical-historiographic research programme capable of highlighting the
consequences one among them can have upon our way of thinking. This way
the continuity aspects can better show the development of the programme, i.e.
when in a revolutionary development it is possible to find some great
scientific or philosophical idea or to search for the antecedents."
He combines two epistemological tools, Kuhn's revolutions and Lakatos'
(1970) methodology of scientific programmes, as clearly outlined in Speranza
(1993a):
"A research programme appears like a sequence of theories (or even like an
evolving theory) with a core containing the fundamental ideas to be defended
at all costs, a negative heuristics to keep the researchers from theory
confutation and a positive heuristics which shows the path to be followed."
According to me, the last mentioned essay outlines Speranza's most complete
point of view. It is a long work which is an organic summary of ideas that
were presented in other texts. Here we can find some ideas he gathered from
two epistemologists who influenced the last phase of his thought, Gonseth
and Bachelard 3. The starting point of this work is to oppose to the current
image of mathematics which is still linked to the platonic-logicist or neonominalist pure rationalism, consequence of Poincaré's conventionalism or
derived from formalism or logic positivism.
He traces all these interpretations back to justification epistemologies and
wishes to oppose to them with more flexible philosophies, the natural
development of what Castellana (1990) calls Italo-French rationalist
epistemology. He recognises Enriques, Bachelard and Gonseth as the
founding fathers of this trend; from Gonseth he gets idoneism, that is the
search for a kind of epistemology suitable for understanding the development
of mathematics, keeping in mind that it is impossible to separate epistemology
of mathematics from that of sciences as science is a single entity. His project
is outlined in the following statement Speranza (1993a):
"This work is meant to contribute to the creation of a non-absolute philosophy
of mathematics (it seems to me suitable to define it with a negative statement
as this is meant to be a non-totalizing philosophy: see Bachelard's philosophie
du non). I wish to draw your attention to the term construction: this
philosophy must not be a given system to which all knowledge is to be
submitted, on the contrary it must be developed upon knowledge and open to
change according to its evolution. This is the meaning of Enriques's (1938)
reappraisal of Kant. I also wish to throw light on some analogies between
fallibilistic philosophies of the early 20th century and the present ones ... .We
shall mainly analyze some features of the above mentioned non-absolute
philosophies; after a conversation with some scholars I had the feeling we
need to clarify some issues."
3
Speranza (1998a) is dedicated to the latter.
87
The first discussed issue is the identification of the epistemology of
mathematics with that of Sciences, because of the major role our discipline
plays in the development of sciences. He then outlines the role of history
especially in connection with the fallibilistic hypothesis, since as Gonseth
says:
"There is no certainty in science independent of the future evolution of
knowledge."
The relationship with history is complicated and Speranza spends a few
essays on it: Speranza (1992b), Speranza (1997a) which is a more complete
version of Speranza (1995b), Speranza (1995a) up to Grugnetti, Speranza
(1999). His position on the matter is very close to Lakatos' for a history of
ideas based on rational reconstructions.
Another issue is the importance of the origin of thought which, according to
him, must be interpreted in two ways: science should throw light on the
creation of thought (Popper); on the other hand the ways of creation of
thought should justify the choices made for the construction and organization
of science. Here he shows his tendency towards (social) constructivism.
More than once he wrote on the quasi-empiricism in mathematics, which he
derived from Lakatos. He mentions some examples of potential istancies of
falsifiability or rather inadequacy of certain formal theories, compared to
informal theories which would be the "ancestors". Later on, he outlines
examples of potential istancies of falsifiability for informal theories in
geometry. However, in Speranza (1993a) he warns:
"I believe it is not possible (or at least inappropriate) to distinguish between
formal and informal theories: elementary geometry à la Hilbert is axiomatic,
not formalised, it is not considered in the language of logic: Tarski (1959) for
example took this further step. However, this can be considered a
formalisation of Euclidean geometry, which is a formalisation of intuitiveexperimental geometry, which is a formalisation of Gonseth's natural science
of elementary truths"
In Speranza (1993a) again he mentions the issue of revolutions and research
programmes in mathematics and he concludes
"I believe non-absolute philosophies to offer good tools for the analysis of
significant moments and aspects of the mathematical thought...
An absolute philosophy leads us to consider as a mistake everything that does
not fit into its domain; non-absolute philosophies on the contrary are more
tolerant, partly owing to the historic sensitivity they are characterised by. If
you consider only the highest levels of science, you doubt, or at least have no
interest in, the majority of people dealing with science (this might be the
origin of the lack of interest of several great scientists in learning and teaching
problems and maybe even of the lack of understanding between scientists and
science philosophers). Non-absolute philosophies revalue interest in all levels
of science and in research at all levels of science. Research is a personal
enrichment, even when it is not completely original: for the non-scientists it is
88
more important to find out something by themselves, though widely known
already, rather than passively walk along somebody else's path".
A tireless curious reader, in the last years of his life Speranza found in
philosophical hermeneutics the right collocation of several ideas he had
published earlier on. His references change to Proclo, Salanskis, Heidegger,
Gadamer, Derrida. His aim is that to prove that an opposition between
‘spiritual sciences’ and ’natural sciences’ is senseless. The examples he
makes in Speranza (1998c) on the translation of the fourth common notion of
Euclid show how the various translators picked different terms according to
their Platonic or Aristotelian position.
From hermeneutics he takes on the concept of hermeneutic circle and
deconstruction. The first concept is widely discussed in a long manuscript,
Speranza (1996) 4. The second one was set as the main issue to be discussed
in the 1999 research programme.
3. Didactics and epistemology.
Didactics is a key element in all Speranza's production, as you can see from
the previous quotation, and it derives from his epistemological approach.
Epistemology and didactics are always deeply intertwined in his works. In
Speranza (1987) he states:
"I wish to conclude with some general reflections on epistemological research
and its impact on didactics. All people working in the didactic field researchers, authors, inservice or training teachers - shall be reminded to be
very careful with epistemological problems. However, it would be impossible
to overestimate the importance of epistemology on the whole."
The above statement is explained in Speranza (1997a)
"Often it is not possible to make a decision on if and how to treat a certain
subject by considering only the technical facts peculiar to mathematical
discourse (the decision that they should be enough itself is epistemological).
Implicit philosophies that influence the mathematical conceptioncan be
different , for example, in teacher and students...
It is important to bring out these implicit philosophies, so as to create a more
rational reference framework for knowledge...
Therefore, the philosophy of mathematics is often necessary to provide a
meaning for the issue. It is better to have a non-absolute philosophy as
reference framework, which cannot exclude the historic dimension. It might
be that the history of teaching comes down to a few single events which give
no ideas for epistemological reflection: it is the consequence of the habit to
leave students with pieces of knowledge to make their own synthesis (it is the
most difficult delicate thing)."
4
It is a very detailed manuscript but it lacks a bibliography, though quoted in the text.
89
He believes even that it is possible to create a virtuous circle where didactics,
history and epistemology support each other and integrate one another, thus
"upsetting" the traditional linear collocation whose origin he traces back to
Comte's basically positivistic classification of disciplines. In Speranza
(1993b) he already treats the problem of the positivist classification of
disciplines applied to Italian University.
In Speranza (1996) he shows how this organisation is
"...a useful idea for the didactic expert, as it allows to organise the
presentation of knowledge in a linear way which is isomorphic to the linear
structure of time."
This metaphor, however is not applicable to teaching because it does not
correspond to the real interaction of disciplines. The idea of a circle is
complex: in Speranza (1996)
"When you have a 'circle' those who need to programme the main lines of
teaching and those who have to perform it are faced with the problem of
analyzing the interaction between the disciplines involved (it is easier to write
a book based on a linear concept rather than a complex one!) ... The main
issue of this essay is that once again we are in a system where we can have the
interaction (not only circular) of psychology, too."
Then he moves on to analyze the 'bilateral relationship' of the various
disciplines and he stresses the fact that the didactic objectives have played a
major role in directing the epistemology of mathematics: some of the basic
texts for the evolution of the mathematical conception had a didactic aim like
Euclid's Elements and Bourbaki's treatise 5. The influence of philosophy on
didactics is even more apparent and the examples brought by Vailati, Loria,
and Enriques are those of people who are directly involved in the
development of didactics, although the notion of didactic research did not
have much meaning at their time. The changes in the paradigms of
epistemology from Hilbert's research programmes, to Lakatos'
falsificationism, through the logical positivism of the Vienna circle,
Bourbaki's implicit epistemology and Popper's critic philosophy, have had
world-wide resonance in mathematics education. The 'official' situation in
Italy was 'frozen' by Gentile's curricula: the effects of the changed paradigms
were felt at first in didactic research and later they were shoved upon Junior
High and Elementary School and upon the reform proposals for High School
curricula.
He believes that since 1980 reserchers in mathematical education have been
faced with the need to lay the basis of didactic research, by looking at
organized philosophical systems; however turning to absolute philosophies
5
90
To these I would add, according to Barnes (1975), the Theory of deductive Science
outlined by Aristotle's Analytica posteriora.
often does not meet the necessary requirements. What is stated in Speranza
(1988) can be interpreted along these lines:
"Allow me to mention some personal memories. About twenty years ago
during my summer holidays I was using my free time to jot down a few
sections of a geometry text for the first year at University. I did not succeed in
completing that work, because I could not find a unitary approach, where to fit
in everything I believed should be part of that book. I was young at the time
and I thought a unitary approach to be the first requirement: with the
experience I gathered in these years, I can say that the composite nature of
teaching was, instead, a positive element. Such nature would have stimulated
a gradual evolution towards other contents: for instance we could leave more
space for the significant applications of abstract algebra."
The philosophy must be suitable (idoine in Gonseth's sense) for didactic
problems, flexible, easily adaptable to the various requirements that
eventually come up (Gonseth's revisibility concept).
The relationship between didactics and history is very rich and important as
well. It is not possible to speak about revolution without the support of history
and even Bachelard's notion of epistemological obstacle has a weaker impact,
since his second motto "... we know against an antecedent knowledge". There
is a short step between epistemological and didactic obstacles, as Bachelard
had already elaborated. The use of these concepts is common in didactics,
especially in French researches. In Speranza (1996) he states:
"All didactic messages are a hint for reflection for teachers; some can even be
more or less explicitly translated in didactic programming and practice ... we
all go through different phases of knowledge as it becomes increasingly
organized and has to face certain difficulties mainly connected to a few basic
steps. If we take the analogy between the development of human knowledge
and that of individual knowledge for granted, we can try to treasure up
history: however, blind faith in history can also be dangerous. It would be a
mistake to present isolated events, on the contrary we need a history capable
of showing the dynamics of the mathematical thought. This is one of the roles
of epistemology: making history 'understood'. epistemology nonetheless has
other functions: explain the 'meaning' of mathematical concepts and theories
(here history comes back again); also help avoiding improprieties and even
real epistemological mistakes which could lead to distorted ways of thinking this is a message to the teachers."
Speranza's thought on the relationship between didactics, history and
epistemology is not a scholar's "curiosity", on the contrary he sees this
virtuous circle as "added value" to be used in practice in class. This is clearly
stated in Speranza & al. (1986) 6:
"What about mathematics ? According to the classical conception it is above
the fray, in the realm of certainty: later on we shall see the sense of this
6
This book was written when a new graduate course for Elementary School teachers
was thought to be underway; it is exhaustive and can be useful for training not only
Elementary School teachers.
91
statement. This notion is of interest for mathematicians who deal with
developed theories; teachers (especially elementary teachers) are interested in
developing theories and even a mathematical theory is experimental in the
beginning. According to Hungarian Imre Lakatos (1922-1974) Popper's
falsifiability principle is valid also for several mathematical theories
(according to Lakatos, the most interesting mathematical theories are
developing theories that are not completed yet).
There has been an objection to Popper's philosophy that it deals mainly with
moments of change or revolution even in science, while in quiet times
scientists work within a framework of accepted knowledge. This objection,
however, cannot concern the learning of science: in fact it is the school's task
to lead the students to their own "spontaneous" view, towards gradually more
organized theories, that are gradually tested. A student has then to walk along
the path of scientific knowledge already traced by humanity in a brief and
simple way. Along this path he will come across several scientific
revolutions".
His interest for didactics was one of the reasons that led Speranza to edit,
together with Lucia Grugnetti, the Italian version of Baruk (1998) - not a
translation - which was published by Zanichelli, Bologna, in December 1998
and which he could not see printed.
References
Works by Francesco Speranza quoted in the text
SPERANZA,, F. and MEDICI, D., QUATTROCCHI, P.: 1986, Insegnare la
Matematica nella scuola elementare. Zanichelli, Bologna.
SPERANZA,, F.: 1987, 'A che cosa serve la filosofia della Matematica?', La
Matematica e la sua didattica, 1987, a. 1, n. 1, 14 - 24 and republished in
Speranza (1997b), 1 - 14.
SPERANZA,, F.: 1988, 'Salviamo la Geometria.', La Matematica e la sua didattica,
1988, 2, n. 2, 6 - 13 and republished in Speranza (1997b), 15 - 24.
SPERANZA,, F.: 1989, 'La razionalizzazione della geometria.' Periodico di
Matematica, VI, 65, 29 - 46 and republished in Speranza (1997b), 25 - 36.
SPERANZA, F.: 1990, 'Il significato filosofico della Matematica e il suo
insegnamento.' Atti del convegno, Il pensiero matematico nella cultura e nella
societ‡ italiana negli anni '90, Quaderni Pristem, n. 1, Documenti, 59 - 66 and
republished in Speranza (1997b), 45 - 50.
SPERANZA, F.: 1992a, 'Tendenze empiriste nella Matematica.', su Speranza, F. and
Ferrari M. (eds.), Epistemologia della Matematica, Seminari 1989 - 1991, prog.
TID - FAIM (n. 10), 77 - 88 and republished in Speranza (1997b), 57 - 64.
SPERANZA, F.: 1992b, 'Il ruolo della storia nella comprensione dello sviluppo della
scienza.', Cultura e scuola, v. 31, n. 123, 201 - 208 and republished in Speranza
(1997b), 79 - 86.
SPERANZA, F.: 1993a, 'Contributi alla costruzione d'una filosofia non assolutista
della Matematica.' Epistemologia, Vol. 16, 255 - 280 and republished in
Speranza (1997b), 87 - 102.
SPERANZA, F.: 1993b, 'La classificazione delle Scienze: un problema concreto con
fondamenti epistemologici.', Rivista di Matematica dell'Università di Parma,
ser. 5, vol. 2, 159 - 170 and republished in Speranza (1997b), 103 - 112.
SPERANZA, F.: 1994a, 'Rivoluzioni in matematica: il caso cartesiano e il caso
bourbakista.' Atti del Convegno SILFS, Lucca, 1993, ETS, Pisa, 128 - 144 and
republished in Speranza (1997b), 151 - 162.
92
SPERANZA, F.: 1994b, 'The influence of some mathematical revolution over
didactical and philosophical paradigms.' on Steiner H.G. & Bazzini L. (Eds):
Proc. of the 2nd Italo-German Symposium on Didactics of Mathematics,
Osnabruck (1992), IDM der Universit‰t Bielefeld, Materielle und Studien
Band 39, 163 - 174.
SPERANZA,, F.: 1994c, 'The Idea of Revolution as an Instrument for the Study of
the Development of Mathematics and for its application to Education.' on
Ernest P. (Ed.): Constructing Mathematical Knowlegde: Epistemology and
Mathematics Education, The Falmer Press, London, 241 - 247.
SPERANZA, F.: 1995a, 'Per il dibattito sulla storia.', Lettera Pristem, n. 18, 8 - 9
and republished in Speranza (1997b), 179 - 180.
SPERANZA, F.: 1995b, 'The Significance of History and of Non-Absolutist
Philosophies of Mathematics in Mathematics Education.', "Perspectives", The
Media and Resources Center, Univ. of Exeter, School of Education, n. 53, 42 51.
SPERANZA, F.: 1996, 'Epistemologia, Storia, Didattica: un circolo virtuoso.'
unpublished manuscript (the date of the used version is only a hypothesis).
SPERANZA, F.: 1997a, 'Il significato della storia e delle filosofie non assolutiste
nella didattica della Matematica.' , Italian extended version of Speranza
(1995b), on Speranza (1997b), 171 - 178.
SPERANZA, F.: 1997b, Scritti di Epistemologia della Matematica, Pitagora,
Bologna. MR 99m:0008 (Otavio Bueno).
SPERANZA, F.: 1998a, 'Rivisitando Gaston Bachelard: la teoria degli ostacoli
epistemologici e la filosofia della matematica', in Abrusci, V.M. & Cellucci, C.
& Cordeschi, R. & Fano, V. (eds.) Prospettive della Logica e della Filosofia
della Scienza, Atti del Convegno Triennale della Società Italiana di Logica e
Filosofia delle Scienze (Roma, 3-5 gennaio 1996), Edizioni ETS, Pisa, 1998,
pp. 453-468.
SPERANZA, F.: (march) 1998b, 'Attenti al neopositivismo logico! ', unpublished
manuscript (the date of the used version is only a hypothesis).
SPERANZA, F.: (september) 1998c, 'Scienza ed Ermeneutica. Un caso esemplare: il
Commento di Proclo al primo libro degli elementi di Euclide.', unpublished
manuscript (the date of the used version is only a hypothesis).
Others authors’works
BARNES J.: 1975, 'Aristotle's Theory of Demonstration' in Barnes J., Schofield M.,
Sorabji R. eds. Articles on Aristotle, 1. Science, Duckworth, London.
BARUK, S.: 1998, Dizionario di matematica elementare, Italian version from the
French edition by Speranza, F. and Grugnetti L., Zanichelli, Bologna.
BOURBAKI, N.: 1939, Éléments de Matématique, Hermann, Paris.
CASTELLANA, M.: 1990, ‘Alle origini della nuova epistemologia’, Il Protagora,
Saggi e ricerche, n. 7, 15 - 100.
DUNMORE, C.: 1992, 'Meta-level revolutions in Mathematics', Gillies, D. (Ed.)
Revolutions in Mathematics, Clarendon Press, Oxford, 209 - 225.
ENRIQUES, F.: 1938, La théorie de la connaissance scientifique de Kant à nos
jours, Hermann, Paris.
ERNEST, P.: 1992, 'Are there revolutions in mathematics', POME Newsletter, nn. 4
- 5.
GILLIES, D. (Ed.): 1992, Revolutions in Mathematics, Clarendon Press, Oxford.
GRUGNETTI, L. and Speranza, F.: 1999, 'General reflections on the problem of
history and didactic of mathematics: Some answers to the Discussion Document
for the ICMI Study on the role of the history of mathematics.', Philosophy of
Mathematics Education Journal, 11, 1999.
KUHN, T.: 1962, The structure of scientific revolution, Chicago University Press,
Chicago.
93
LAKATOS, I.: 1967, ‘A rennaissance of empiricism in the recent philosphy of
mathematics?’, in Lakatos, I. Philosophical Papers, vol. 2, Cambridge U.P.,
1978, 24 - 42.
LAKATOS, I.: 1970, ‘Falsification and the methodology of of scientific research
programmes’, in Lakatos, I. Philosophical Papers, vol. 1, Cambridge U.P.,
1978, 8 - 111.
LAKATOS, I.: 1971, ‘History of science and its rationale recinstruction.’ in Lakatos,
I. Philosophical Papers, vol. 1, Cambridge U.P., 1978, 102 - 138.
POPPER, K.R.: 1970, La logica della scoperta scientifica, Einaudi, Torino.
TARSKI, A.: 1959, ‘What is elementary geometry?’, in Henkin, l. & Suppes, P. &
Tarki, A. (eds.) The axiomatic method with special reference to geometry and
physics, North Holland, Amsterdam, 16 - 29.
94
95
PART TWO
96
97
KEYS FOR CLASSIFICATION OF ABSTRACTS
In order to give an outline of the orientation of the research we have asked the
authors to classify their papers according to the following streams:
School level
c
e
m
b
t
u
= Infant school
= Elementary school
= Lower secondary school
= Initial two yeras of upper secondary school
= Last three years of upper secondary school
= university level.
Mathematical subject
a
ar
c
g
hs
l
p
s
= Algebra
= Arithmetic
= Calculus
= Geometry
= History of Mathematics
= Logic
= Probability
= Statistic.
Educational area
ap
cm
cr
d
i
m
mr
p
pcb
ps
tcb
tr
tt
v
va
98
= applications; as = Astronomy
= Computer and Mathematics
= Curriculum Research
= Didactics; e = epistemology
= Image of mathematics
= Metacognition, social and affettive factors
= Models and representations
= Proofs
= Pupils Beliefs and Conceptions
= Problem Solving
= Teachers Beliefs and Conceptions
= Theoretical educational Research
= Teacher Training
= Visualization
= Evaluation
99
A SURVEY OF THE ITALIAN PRESENT RESEARCH
IN MATHEMATICS EDUCATION
ACCASCINA G., BERNESCHI P., BORNORONI S., DE VITA M., DELLA ROCCA
G., OLIVIERI G., PARODI G.P., ROHR F.
La strage degli innocenti. Problemi di raccordo in matematica tra scuola e
università (The slaughter of the Innocents. Bridging problems in mathematics
between high school and university). Centro di Ricerche Didattiche Ugo
Morin, Giovanni Battagin Editore, Bassano del Grappa, 1998
school level: t, u; mathematical subject: m; educational area: cr, tcb.
A study of the slaughter of students in their first year at university when
taking mathematics exams. The percentage of failures in exams of the first
year mathematics courses is very high.
The aim of the research was to measure not only the real basic knowledge in
mathematics but also what their schoolteachers and university professors
presumed they knew. The research is based on the analysis of a set of
questionnaires filled in by mathematics school teachers, university professors
and first year students on the university degree courses in mathematics and
computer science and on structured interviews with the students. It emerges
that: (1) first year students have a poor grasp of mathematics; (2) teachers and
professors have not fully understood how weak students are in mathematics.
Furthermore it emerges that first year students do not seem realise the
difficulties that they are going to face. They understand them only after their
first exams. Students, until their first exams, are innocent, in the sense that
they know nothing of evil. This explains the title ‘The slaughter of the
innocents’.
AGLÌ F., D’AMORE B., MARTINI A., SANDRI P.
Attualità dell’ipotesi “intra-, inter-, trans-figurale” di Piaget e Garcia,
L’Insegnamento della Matematica e delle Scienze Integrate, 1997, 20A, 4,
329-361.
school level: c, e; mathematical subject: g; educational area: mr.
Research problem and results: In this paper we present some tests we have
carried out in several Italian cities, in order to verify experimentally the
“intra-, inter-, trans-figural” hypothesis by J. Piaget and R. Garcia,
exclusively in the field of geometrical figure. The results seem to contradict
some statements of those Authors, and particularly the assumed necessity of
the hierarchical scale of those three levels.
School-level: last class of pre-primary school (pupils aged 5-6 years) and 1st
class of primary school (pupils aged 6-7 years).
ANDRIANI M.F., FOGLIA S.
Comici Spaventati Matematici (Funny, Frightened Mathematicians), in I.
Aschieri, M. Pertichino, P. Sandri, P. Vighi (Eds.), Atti del Convegno
Nazionale “Matematica e difficoltá” n. 7, Matematica e affettivitá. Chi ha
paura della matematica?, 1998, Pitagora, Bologna, 101-106.
school level: b, t; mathematical subject: m; educational area: i.
Posing the unexpected question to students aged between 14 and 19 attending
Middle High Schools: “what is Maths?” This is the subject of the survey
carried out in Parma between the end of 1996 and the first months of 1997: its
results are surprising. Mathematics seems to touch the students’ most
sensitive chords as something intimate, private and deep. From the answers
received, we can determine different profiles that can be divided into distinct
groups: the forced: those compelled to study it against their will; the
secessionists: those who just absorb the most digestible parts; the advers:
those who are irritated, deem it useless, exausting, not too much pertaining to
reality, a iumble of absurd formulas and distressing problems; the possibilists:
they are pacefully disinterested but willing to study it enough to pass their
exam; the convinced supporters: the few one who love it; those who say ...it
depends on the teacher: those who get involved depending their teacher’s
ability.
ANDRIANI M.F., GRUGNETTI L.
Nature interactive des problèmes non standard (The interactive nature of nonstandard problems), in P. Abrantes et al. (eds) The interactions in the
mathematics classroom, Proc. CIEAEM 49, (Setubal, Portugal, 1997), 1998,
280-285.
school level: m, b; mathematical subject: g; educational area: d.
In this paper, concerning a problem solving research, two kinds of
interactions in the mathematics classroom are discussed: the interaction
between pupils (12-15 years) and the interaction between the pupil and a
problem. The authors define a problem as a new activity which is meaningful
to the students and which must be sufficiently clear to their current
knowledge to be assimilated and yet must be sufficiently different in order to
force them to transform their methods of thinking and working. In this paper,
the focus is on both the quality of problems and pupils' reactions and
interactions on order to learn mathematics.
100
ANDRIANI M.F., DALLA NOCE S., GRUGNETTI L., MOLINARI F., RIZZA A.
Autour du concept de limite (On the concept of limit), in F. Jaquet (ed.)
Relationships between classroom practice and research in mathematics
education, Proc. CIEAEM 50, (Neuchatel, Switzerland, 1998)1999, 329-335.
school level: b, t; mathematical subject: ac;
educational area: pcb, e,
d.
The paper deals with research about difficulties related to the learning of the
concept of limit. Some students' interpretations of the term "limit" and of the
term "infinity" are presented and analysed in connection with their difficulties
concerning the use of the concept of limit in successions and in functions.
The strong power of the interpretations of limit as "a barrier" and of infinity
as "something that has no limit" on mathematics questions are studied by the
analysis of students' answers to questions concerning discrete and continous
problems.
ARCHETTI A., ARMIENTO S., BASILE E.,CANNIZZARO L., CROCINI P.,
SALTARELLI L.
Influenza della sequenza delle informazioni nella risoluzione di un problema
(The influence of the sequence of data in problem solving), L'Insegnamento
della matematica e delle Scienze Integrate, 2000, vol. 23A, n.1, 7-26
school level: e, m; mathematical subject: ar; educational area: ap.
Our research question was about relations between difficulties in solving
word problems and time structures in the stories used and our purpose was to
provide evidence that students prefer situations in which the order of
informations given in the word text of the problem (Time Text - TT)
corresponds to the order in which the action evolves in real life
(Chronological Time - CT), much more than situations where TT corresponds
to the order in which numerical data have to be treated in the resolution
procedure (Solution Time - ST). Five problems were posed in 3 linguistic
formats (to agree with different time structures) to 171 children aged 11 to 12.
Children had been graded within four problem solving ability levels and then
distributed into 3 homogeneous groups. Each group was presented with all the
Time structures in different problems. Empirical data show that students of
any ability level do fare better when the order of information provided in text
corresponds to the order of actions in real life situations, as well as clues
(revealing or misleading) can be provided by means of rearrangement of
problem text.
101
ARDIZZONE M.R., CILENTO E., LANCIANO N., MARLIA A.M., PIEROTTI A.
Dal Pantheon alla geometria - una ricerca che è anche ricerca di innovazione
(From Pantheon to geometry - an innovation-oriented research),
L'Insegnamento della matematica e delle Scienze Integrate, 1994, n.1
school level: e, m; mathematical subject: g; educational area: pcb.
In this paper we present an investigation about the relationship between
children and real space, in primary school (6-13 years old). Our position, our
vision, our perception, our feeling in real space is the beginning of geometry.
We present our teaching hypothesis and the actions proposed to explore a big
space: the Pantheon in Rome. The research continues with Lanciano N. et al.
'Lecture d'un espace urbain' ('Interpretation of an urban space'), XLVI
Rencontre de la CIEAEM, Toulouse 1994, and a book: Lanciano N. et al. La
Geometria in città, Centro Morin - G.Battagin
ARRIGO G., D’AMORE B.
“I see it but I don’t believe it”. Epistemological and didactic obstacles to the
process of comprehension of a theorem of Cantor that involves actual infinity,
Scientia Paedagogica Experimentalis (Gent, Belgium), XXXVI, 1, 93-120.
Spanish translation in press in Educacion matematica (Mexico DF, Mexico);
Italian translation in press in L’Insegnamento della Matematica e delle
Scienze Integrate (Paderno, Italy).
See also ARRIGO G., D’AMORE B, An ample summary in English:
Epistemological and didactic obstacles to the process of comprehension of a
theorem of Cantor that involves actual infinity, Proceedings of Osnabrück,
CERME 1, 1998, in print.
school level: e, m, b, t; mathematical subject: m; educational area: mr.
Main theoretical frame: Our work is mainly referred to two problems: the
theory of obstacles and the learning of infinity (with its specific difficulties).
As regards international literature, we made particular reference to works by
G. Brousseau, R. Douady, R. Duval, E. Fischbein & Alii, L. Moreno & G.
Waldegg, G. Shama & B. Movshovitz Hadar, R. Stavy & B. Berkovitz, D.
Tall, P. Tsamir & D. Tirosh and other Authors.
Research problem: In this work we study the limits of comprehension and
acceptance on the part of students in the upper secondary school, in relation to
some recent questions as to the actual use of infinity and in particular about a
celebrated theorem of George Cantor. We attempted moreover an analysis of
the motivation of this widespread non-acceptance, collating it in various
ways.
102
School-level: last two years of high school in Italy and in Switzerland (pupils
aged 17-19 years).
In the same field, see:
D’AMORE B. (1996). El infinito: una historia de conflictos, de sorpresas, de
dudas. Un campo fértil para la investigación en didáctica de la matemática.
Epsilon (Sevilla, Spain), 36, 341-360.
Italian translation: L'infinito: storia di conflitti, di sorprese, di dubbi. Un
fertile campo per la ricerca in Didattica della Matematica, La Matematica e la
sua didattica (Bologna, Italy), 3, 1996, 322-335 (I); 3, 1997, 289-305 (II).
This text was the opening and final lecture of ICME 8, Sevilla (Spain), 14-21
July 1996, Topic Group XIV (Infinite processes throught the curriculum), in
which the Author was Chief Organizer, and Raymond Duval (France) and
Vera W. De Spinadel (Argentina) were Advisory Panels.
ARZARELLO F., BARTOLINI BUSSI M. G.
Italian Trends in Research in Mathematics Education: A National Case Study
in the International Perspective, in Kilpatrick J. & Sierpinska A. (eds.),
Mathematics Education as a Research Domain: A Search for Identity, 1998,
vol. 2, 243-262, Kluwer Academic Publishers.
school level: c, e, m, b, t, u;
mathematical subject: m; educational area: tr.
Every discussion about Research in Mathematics Education seems to
emphasize the existence of different research traditions that are developed
locally with their own store of epistemological debates, institutional
constraints, research questions, methods, results and criteria, often leading to
the birth of specific paradigms. Despite this increasing volume of information
at the international level, the discussion of research questions related to local
projects seems to be difficult, because of problems of communication and of
relevance. This paper aims to address both sets of problems by starting from
an analysis of the Italian situation. In the first part of the paper the authors
present some elements of a national case study, in order to communicate
information about the roots and the present state of the core of Italian research
in mathematics education. In the second part of the paper, the authors,
drawing from the national case study, pose some tentative implications for the
international professional community of researchers in mathematics
education; above all its contribution to the emergence of an original paradigm
that has at its core research for innovation.
ARZARELLO F., BAZZINI L., CHIAPPINI G.
103
L'Algebra come strumento di pensiero: analisi teorica e considerazioni
didattiche (Algebra as a tool for thinking: theoretical analysis and didactical
remarks), CNR-TID project, FMI series, 1994, vol. 6.
school level: b; mathematical subject: al; educational area: mr.
The major focus is on a theoretical model apt to interpret the underlying
dynamics of algebraic thinking. This model applies the Frege’s semiotic
triangle to the analysis of symbolic expressions, which are seen as a
combination of the triad “Sign, Sense and Denotation”. Consequently,
algebraic thinking (and learning) emerges as a “game of interpretation”, and
the construction of mathematical knowledge grows in the social negotiation
of meanings used by pupils in solving problems.
The book is subdivided in three parts. The first part frames the research study
in the context of existing literature. The second part describes the theoretical
model. Finally, the third part approaches some didactical implications, in the
light of the above mentioned paradigm. Previous work of the same authors,
dealing with a theoretical frame which uses key -concepts taken from logic
and linguistics, is described in:
ARZARELLO F., BAZZINI L., CHIAPPINI G.P., 1994 'Intensional semantics as
a tool to analyze algebraic thinking', Rendiconti del Seminario Matematico,
Vol.52, n.2, (105-125).
ARZARELLO F., BAZZINI L., CHIAPPINI G.P.
'The process of naming in algebraic problem solving', Proc. PME 18, Lisbon,
Portugal, 1994, vol. 2, 40-47.
school level: b; mathematical subject: al; educational area: mr.
This study is located in a wider analysis of the nature of algebraic language
and its related cognitive processes, which has been carrying out by the
authors. This analysis points out the difference between “sense” and
“denotation” of a symbolic expression as a key element of algebraic thinking
in problem solving: this distinction and the mutual interaction of the two
poles are investigated. Particularly, this report focuses on the relationship
between sense and denotation of a symbolic expression during the process of
naming, that is the process of assigning names to the elements of a given
problem. This study provides an analysis of the process of naming and of the
difficulties met by a sample of undergraduated students and a sample of
secondary school students.; evidence is given about the crucial role of the
naming process in determining the success of the solution.
ARZARELLO F., BAZZINI L., CHIAPPINI G.
104
The construction of algebraic knowledge: towards a socio-cultural theory and
practice, Proc. PME19, Recife, Brazil, 1995, Vol 1 , 119-134.
school level: m; mathematical subject: al; educational area: d.
This study starts from the theoretical framework elaborated by the authors in
previous work , on the basis of students’ observed behaviors while solving
algebraic problems. In such a framework algebra is considered as a language
and as a thinking tool. Here the authors feature the environment where
algebraic thinking finds its proper place. In particular the social space of a
subject and the didactic space-time of production and communication are
analysed and grounded in teaching activities for primary and secondary level
(approximately age 10-14).
In particular, the processes of anticipation and planning are taken into account
and framed in activities which concern producing-transforming-interpreting
algebraic expressions.
ARZARELLO F., GALLINO G., MICHELETTI C., OLIVERO F., PAOLA D.,
ROBUTTI O.
Dragging in Cabri and modalities of transition from conjectures to proofs in
geometry, in Proc. PME 22, Stellenbosch, South Africa, 1998, vol. 2, 32-39.
school level: b, t; mathematical subject: g; educational area: p.
In this report we analyse some modalities that feature the delicate transition
from exploring to conjecturing and proving in Cabri: we use a theoretical
model that works in other environments too. We find that the different
modalities of dragging are crucial for determining a productive shift to a more
'formal' approach. We classify such modalities and use them to describe
processes of solution in Cabri setting, comparing it with the pencil and paper
ones.
ARZARELLO F., MICHELETTI C., OLIVERO F., PAOLA D., ROBUTTI O.
A model for analysing the transition to formal proofs in geometry. in Proc.
PME 22, Stellenbosch, South Africa, 1998, vol. 2, 24-31.
school level: u; mathematical subject: g; educational area: p
This report sketches a model for interpreting the processes of exploring
geometric situations, formulating conjectures and possibly proving them. It
underlines an essential continuity of thought which rules the successful
transition from the conjecturing phase to the proving one, through exploration
and suitable heuristics. The essential points are the different type of control of
the subject with respect to the situation, namely ascending vs. descending and
105
the switching from one to the other. Its main didactic consequence consists in
the change that the control provokes on the relationships among geometrical
objects. The report relates the research to the existing literature (§1) and
exposes the main points of the model through the analysis of a paradigmatic
case (§§ 2,3); in the end (§4) some partial conclusions are drawn.
ARZARELLO F., OLIVERO F., PAOLA D., ROBUTTI O.
Dalle congetture alle dimostrazioni. Una possibile continuità cognitiva,
L’Insegnamento della Matematica e delle Scienze Integrate, 1999, vol. 22B,
n.3.
school level: b, t; mathematical subject: g educational area: p.
The research that we are carrying out suggests that there is an essential
continuity of thought which rules the successful transition from the
conjecturing phase to the proving one, through exploration and suitable
heuristics. The essential points are the different type of control of the subject
with respect to the situation, namely ascending vs. descending and the
switching from one to the other. Its main didactic consequence consists of the
change that the control provokes on the relationships among geometrical
objects. Ours findings are that Cabri géomètre strongly helps the transition
from one type of control to the other. In particular we found out the different
modalities of dragging are crucial for determining a productive shift to a more
‘formal’ approach. In this paper we outline the different modalities of
reasoning (paragraph 2) and of dragging (paragraph 3) which we observed in
processes of problem solving, either in the phases of productions of
conjectures, or in the phases of their validation. Then we analyse the protocol
of a pair of students which are engaged in solving a geometry problem in
Cabri (paragraph 4).
ARZARELLO F., OLIVERO F., PAOLA D., ROBUTTI O.
I problemi di costruzione geometrica con l’aiuto di Cabri, L’Insegnamento
della Matematica e delle Scienze Integrate, 1999, 22B, n .4.
school level: b, t; mathematical subject: g; educational area: p
Construction problems play an important role in the transition to proof, as far
as their theoretical nature is concerned. In this paper we discuss the results of
our ongoing research about students’ approach to proofs through construction
problems; this construction activity involves the use of the microworld CabriGéomètre, which helps pupils to find the correct sequence of menu commands
to produce a figure that can be validated through the dragging test, as well as
106
in the traditional paper and pencil environment, in which correct procedure
must find a mathematical justification.
AURICCHIO V., DETTORI G., GRECO S., LEMUT E.
Learning to bridge classroom and lab activities in math education, in
D.Passey & B.Samways (Eds.) Information Technology: Supporting change
through teacher education, Chapman&Hall, 1997.
school level: b; mathematical subject: m; educational area: cm, tt.
When introducing computer laboratories in mathematics education, some
training for in-service teachers appears essential to help them revise school
programs and give an answer to issues related with teaching habits, didactical
planning and the conduct of work in the mathematics laboratory. In order to
make computers an effective support to mathematics teaching, we think that
class and laboratory should be given equal cognitive importance. Training
courses should be hands-on, give models rather than recipes, and include the
formation of working groups connected to the research world.
BAGNI G. T.
Limite e visualizzazione: una ricerca sperimentale, L’Insegnamento della
Matematica e delle Scienze Integrate, 2000
school level: t;
mathematical subject: c; educational area: cr, d, v.
In this paper the concept of limit in the learning of mathematics is
investigated in Italian High School (pupils aged 18-19 years): the status of the
concept is studied by two tests, particularly referred to the investigation of the
role of visualization (see works by I. Dimarakis & A. Gagatsis, R, Duval, E.
Fischbein, A. Sfard, and: G.T. Bagni, Visualization and Didactics of
Mathematics in High School: an experimental research, Scientia Paedagogica
Experimentalis, 35, 1/1998; G.T. Bagni, The influence of texts’ mental
images upon problems’ resolutions, Proceedings of 2nd Mediterranean
Conference on Mathematics Education, Nicosia, Chyprus, 2000). We
conclude that the visual representation of some infinitesimal methods is
tacitly considered by pupils in the sense of potential infinitesimal, and that an
improper use of visual methods may be quite ineffective for the correct
learning of the limit. Moreover we underline that a correct introduction of the
limit concept in the sense of actual infinitesimal can really help the students to
overcome some dangerous misconceptions.
BAGNI G. T., D’ARGENZIO M. P., RIGATTI LUCHINI S.
107
A paradox of Probability: an experimental educational research in Italian
High School, Proc. of the International Conference on Mathematics
Education into the 21st Century: Societal Challenges, Issues and Approaches,
Cairo, Egypt,1999, III, 57-61.
school level: t;
mathematical subject: p; educational area: d, pcb.
An informal point of view can be important and interesting in order to
introduce the concept of Probability. In this paper we describe an
experimental research activity about a first approach to Probability: we
presented to students aged 16-17 years a short test based upon a well known
paradox. The greater part of the pupils considered by intuition Laplace
definition and applied it, but sometimes they made errors and mistakes and
we consider that these errors are also caused by affective elements.
BALZANO E., MELONE N., MORELLI A., RUSSO E., SASSI E., TORTORA, R.
GeT, Software didattico per la Geometria (GeT, educational software for
Geometry) - Atti del Convegno "Didamatica 94" , 1994, Cesena.
school level: b; mathematical subject: g; educational area: cm.
The paper contains the description of the educational software package GeT
(Geometry and Transformations). It is designed to encourage 'open strategies'
and thus can be used in different contexts and with different goals. It is
particularly useful for developing a Geometry curriculum in lower high
school, and it is possible to use it in the following three years. GeT is the
result of a joint venture between the Italian National Education Board and the
Department of Mathematics of the University of Naples and have been
developed by a group of experts in mathematics, physics, computer science
and logic.
BARDONE L., LANZI E., PESCI A.
Una definizione di rettangolo con la mediazione di Cabri in quarta elementare
(The definition of rectangle through the mediation of Cabri in the fourth class
of primary school), L’insegnamento della Matematica e delle Scienze
Integrate, 1998, Vol. 21A, n. 1, 29-52.
school level: e;
mathematical subject: g; educational area: cm, cr.
The article deals with an experience carried out in the frame of geometry with
10-11 year old pupils. The didactical plan has been developed to explore the
hypothesis that the use of Cabri - Géomètre (completed by appropriate
activities with paper and pencil and well organised discussions) may provide
a contribution to the development of pupils’ geometrical thought. In
108
particular, the formulation of a specific definition of rectangle was the main
objective reached by pupils. By exploiting the dynamic aspects of Cabri Géomètre, the interesting and demanding of necessary and sufficient
conditions for defining geometrical figures have been addressed, even in the
first steps of the didactic plan.
The whole experience is summarized and the original work sheets for pupils
are presented togeter with the most significant pupils’ protocols.
BARTOLINI BUSSI M. G.
Analysis of Classroom Interaction Discourse from a Vygotskian Perspective,
in Meira L. & Carraher D. (eds.), Proc. PME 19, Recife, Brazil,1995, vol. 1,
95-101.
school level: e;
mathematical subject: ar, al; educational area: m, mr.
This paper is related to the participation in the plenary panel held in PME
19th in Recife. In the panel thirteen minutes of video footage were used to
focus discussion around a common set of data. The data were presented by
means of videotape from a 5th grade public school classroom in Cambridge,
MA, where students were using an instructional device, which simulated
diving activity with two small puppets. The students had to collect data and
construct and interpret number tables containing these data. The analysis
concerns the adult role in the interaction, by distinguishing a) the macro-level
(how adults set the stage, where interaction was to happen); b) the micro-level
(how adults took part in the interaction).
BARTOLINI BUSSI M. G.
Mathematical Discussion and Perspective Drawing in Primary School,
Educational Studies in Mathematics, 1996, 31 (1-2), 11-41.
school level: e;
mathematical subject: g; educational area: m, cr, p.
The aim of this paper is to analyse the functions of semiotic mediation in a
long term teaching experiment on the plane representation of threedimensional space by means of perspective drawing, that has been carried out
from grade 2 to grade 5 in three different classrooms within the research
project Mathematical Discussion. On the one hand, drawing has a functional
role in the overall development of the child while, on the other, perspective
drawing has a phenomenological role in the genesis of modern geometry. The
experiment aims at connecting (1) pupils' spatial experiences to the
development of the geometry of three-dimensional space and (2) pupils'
drawing experiences to the geometry of two-dimensional space, up to the
mastery of early geometrical strategies of plane representation of space.
Classroom activity alternates with individual problems and classroom
109
discussions orchestrated by the teacher. After a brief introduction containing
some contextual information the problem of the social construction of
knowledge is addressed and some theoretical constructs mainly borrowed
from the Vygotskian school are elaborated; then two analyses of the
experiment are made, according to the motives of activity and to the sequence
of actions; finally the role of semiotic mediation in the whole experiment is
analysed; in the final section some results are recapitulated and compared
with the literature on the teaching and learning of geometry, while the
function of semiotic mediation is discussed with reference to the other
distinctive features of the teaching experiment.
BARTOLINI BUSSI M. G.
Drawing Instruments: Theories and Practices from History to Didactics,
Documenta Mathematica - Extra Volume ICM, 1998, vol. 3, 735-746.
school level: b, t, u;
mathematical subject: g, hs;
educational area: p.
This paper concerns an invited 45-minutes lecture in Section 18: ‘Teaching
and Popularization of Mathematics’ of the International Congress of
Mathematicians (ICM-98), Berlin, August 1998. The paper offers the outline
of a collective research project about proof in geometry developed by some
Italian research groups (directed by Arzarello, Boero and Mariotti, beside the
author). The aim is to show by means of a paradigmatic example how
different companion disciplines (such as epistemology, history, psychology,
sociology) complement didactics of mathematics, by offering analytical tools
to produce results that can increase the knowledge of the teaching and
learning processes in the classroom, produce effective innovation in schools
and, at a larger level, influence the development of school systems. The
paradigmatic example concerns activity with linkages and other drawing
instruments at secondary and university levels. The main thesis of the lecture
is the following: By exploring linkages and other drawing instruments with
suitable tasks under the teacher's guidance, secondary and university students
can: 1) be enriched with a well-balanced image of mathematics, where
theoretical aspects and applications are strictly intertwined, yet not confused;
2) be involved in the generation of 'new' (for the learners) pieces of
mathematical knowledge by taking an active part in the production of
statements and the construction of proofs in a reference theory, living an
experience similar to the one of professional mathematicians; 3) be
acquainted with a set of exploring strategies and representative tools that
nurture the creative process of statement production and proof construction
and are transferable to other sets of problems.
Other references:
110
BARTOLINI BUSSI M. G., NASI D., MARTINEZ A., PERGOLA M., ZANOLI C.,
TURRINI M. & al, 1999, Laboratorio di Matematica: Theatrum Machinarum,
1° CD rom del Museo, Modena: Museo Universitario di Storia Naturale e
della Strumentazione Scientifica <http: //www.museo.unimo.it/theatrum>.
BARTOLINI BUSSI M. G. & MARIOTTI M. A. (1999), Instruments for
Perspective Drawing: Historic, Epistemological and Didactic Issues, in
Goldschmidt G., Porter W. & Ozkar M. (eds.), Proc. of the 4th International
Design Thinking Research Symposium on Design Representation, III 175185, Massachusetts Institute of Technology & Technion - Israel Institute of
Technology.
BARTOLINI BUSSI M. G.
Joint Activity in the Mathematics Classroom: a Vygotskian Analysis, in
Seeger F., Voigt J. & Waschesho U. (eds), The Culture of the Mathematics
Classroom. Analyses and Changes, Cambridge University Press, 1998, 13-49.
school level: e, m; mathematical subject: g; educational area: m.
The paper addresses the issues of the quality of social interaction and the
quality of educational contexts in the mathematics classroom. The first part is
a detailed presentation of the theoretical construct of mathematical discussion,
developed by the author in a Vygotskian and Bachtinian perspective, meant as
a polyphony of articulated voices on a mathematical object (either concept or
problem or procedure or belief) that is one of the motives of the teaching
learning activity: beside the scripts of the main types of mathematical
discussion, several communicative strategies are described. In the second part
of the paper a comparison between different context is started, with a special
emphasis on the problem of coordination and conflicts between them.
Other references:
BARTOLINI BUSSI M. G., 1998, Verbal Interaction in Mathematics
Classroom: a Vygotskian Analysis, in Steinbring H., Bartolini Bussi M. &
Sierpinska A. (eds), Language and Communication in the Mathematics
Classroom, Reston VA: NCTM, 65-84.
BARTOLINI BUSSI M. G., BOERO P.
Teaching Learning Geometry in Contexts, in Mammana C. & Villani V.
(eds.), Perspectives on the Teaching of Geometry for the XXI Century, 1998,
52-61, Kluwer Academic Publishers.
school level: e, m, b, t; mathematical subject: g; educational area: ap.
The reference to 'real' contexts in teaching-learning geometry (and more
generally mathematics) has been and still is widespread among mathematics
111
educators in this century. The authors propose a theoretical framework which
includes cultural and cognitive issues involved in teaching-learning geometry
in contexts, and a related classification of contexts and geometrical activities
that are developed within them. The report draws on several case studies of
teaching-learning geometry in contexts, that have been developed by research
teams in Genoa and in Modena, for different school levels (elementary and
secondary school). The main examples concern: sunshadows; representation
of the visible world by means of perspective drawing; mathematical
machines. i. e. linkages and kinematic geometry.
BARTOLINI BUSSI M. G., MARIOTTI M. A.
From Drawing to Construction in the Cabri Environment: the Role of Teacher
Intervention, in Proc. PME 22, Stellenbosch, South Africa,1998, vol. 2, 6471.
school level: b; mathematical subject: g; educational area: cm, m, p.
Referring to a long-term experimental project focused on the introduction to
mathematical proof, this paper presents the analysis of a collective discussion,
taking place in a 9th grade class. The discussion deals with different strategies
for constructing a square in the Cabri environment. The analysis has two
objectives. On the one hand, to show the evolution of the justification process
centered on the shift from checking on the product to checking on the
procedure. On the other hand, to show how the relation to drawing is
modified by the mediation of the Cabri environment as the teacher
accomplishes it.
BARTOLINI BUSSI M. G., MARIOTTI M. A.
Semiotic mediation: from history to mathematics classroom, For the Learning
of Mathematics, 1999, 19 (2), 27-35.
school level: u; mathematical subject: g; educational area: pcb, v.
The report starts from the cognitive analysis of an imaginary debate reconstructed with excerpts from historical sources - concerning the shapes of
particular sections of a right cone and of a right cylinder. The analysis, based
on the theory of figural concepts, suggests the following hypothesis: When
conic sections are concerned, a break between the figural and the conceptual
aspects is expected and is not easy to be overcome. An exploratory study with
expert university students was carried out to validate the hypothesis and to
find also what kind of conceptual control, if any, were students able to
mobilize in order to overcome the break. After reporting the findings of the
112
study, we analyse the tools of semiotic mediation introduced in order to help
the students acquire the conceptual control which they lack.
BARTOLINI BUSSI M. G., BONI M., FERRI F., GARUTI R.
Early Approach To Theoretical Thinking: Gears in Primary School,
Educational Studies in Mathematics, 39, 66-87.
school level: e;
mathematical subject: g; educational area: ap, m, p.
Gears are part of everyday experience from very early childhood. This paper
analyses a teaching experiment conducted with 4th graders in the field of
experience of gears. The aim is to identify the characteristics which, given a
suitable sequence of tasks and proper teacher guidance, have enabled the
pupils to approach theoretical thinking, and in particular mathematical
theorems. The authors have focused on the relationships between the
epistemological analysis of some pieces of mathematical knowledge brought
into play in tasks concerning gears, cognitive analysis of pupil construction of
those pieces of mathematical knowledge, and didactic analysis of the
teacher’s role in designing tasks and in offering cultural mediation. This paper
presents the early findings of the teaching experiments, both at the external
level of interpersonal classroom processes and at the inner level of individual
mental processes.
Other references:
BARTOLINI BUSSI M. G., BONI M., FERRI F., GARUTI R. (1998), Wheels and
Circles: Teacher's Orchestration of Polyphony, The Interaction in the
Mathematics Classroom (Proc. CIEAEM 49), 324-332, Setùbal (Portugal).
BASSO M., BONOTTO C., SORZIO P.
Children’s understanding of the decimal numbers through the use of the ruler,
Proc. PME 22, Stellenbosh, South Africa, 1998, Vol. 2, 72-79.
school level: e;
mathematical subject: ar; educational area: cr.
This is an exploratory study about the use of ruler, familiar tool and also
cultural artifact (see Saxe, 1991), to introduce the concept of decimal number,
in the normal classroom curriculum, with third-grade children. We propose
that the children’s use of the ruler can have a mediational role to enable
children to construct their understanding of decimal numbers because it
'externalizes', makes relevant the number line image-schema, which seems a
very effective mental representation (see McClain & Cobb, 1996).
Furthermore the rule can have a mediational role in their understanding of the
additive structure underlying the standard written decimal notation. In order
to achieve our objective, we have designed a classroom practice that engages
113
students in a sustained mathematical activity which requires an extensive use
of the ruler to accomplish different functions (measuring, drawing segments,
ordering and approximating decimal numbers). Opportunities and constraints
in children’s use of the ruler to achieve the educational goal are presented.
Further researches about measurements and the use of measuring instruments
were carried out in fifth grade classes; the results of these investigations can
be found in
BONOTTO, C., MADDALOSSO, M.:1997, Problematiche emerse dall’analisi di
indagini sulla misura, Atti del 2° Internuclei Scuola dell’obbligo, Università
degli Studi di Parma, 59-63.
BAZZINI L.
Sulla comprensione del concetto di funzione in studenti di liceo Scientifico
(On the understanding of the concept of function by high school students), in
Piochi B. (ed.) Funzioni, Limiti, Derivate, come, perchè, quando, con quali
strumenti insegnare l'Analisi nei diversi ordini di scuola. ATTI del IV
Convegno Internuclei per la Scuola Superiore, IRRSAE Toscana, 1994, (3340).
school level: t;
mathematical subject: al; educational area: pcb.
This report is part of an ongoing study concerning the students difficulties
when dealing with the concept of function. Here the focus is on the results of
a questionnaire submitted to a sample of high school students, aimed at
testing their comprehension and mastery of the definition of function and the
notion of equality for two given functions. Such notions have been previously
given in formal terms, referring to relations between sets. The data analysis
shows clearly a lack of mastery in the definition of the concept of function as
well as in the equality of functions. As a consequence, the author challenges
the practice of giving formal definitions at a relatively early stage, preventing
a balanced development which takes into account both operational and
relational aspects.
BAZZINI L. (ed)
Theory and Practice in Mathematics Education. Proceedings of the Fifth
International Conference on Systematic Cooperation between Theory and
Practice in mathematics Education, Grado (GO), 23-27 May, 1994, ISDAF,
Pavia.
school level: -;
mathematical subject: m ;
educational area: cr, tt, e.
This book is a collection of papers presented at the Fifth SCTP Conference,
held in Grado in 1994.Specific questions related to the general theme of the
interaction between theory and practice are approached here.
114
The focus has been on the following topics:
The role played by classroom observation in mediating theory and practice;
The role played by teachers' and students' beliefs in mediating theory and
practice;
The role played by theory and practice in designing curricular materials for
students or for teachers;
Implications for research methodologies in mediating theory and practice.
The book includes contributions by B. Andelfinger, C.Batanero et al., A.Bell,
P.Bero, P.Boero et al., R. Borasi et al., L.Burton, T.J.Cooney, F. Furinghetti,
G. Gjone, L.Grugnetti, F. Jaquet, K. Krainer, S. Lerman, E. Love, N.A.
Malara, J.P. Ponte, J. Radnai-Szendrei, L. Rogers, N. Rouche, F. Seeger, A.
Sierpinska, C. Vicentini.
BAZZINI L.
Il pensiero analogico nell'apprendimento della matematica: considerazioni
teoriche e didattiche' ('Analogic thinking in Mathematics learning: theoretical
and didactical remarks'), L'Insegnamento della matematica e delle Scienze
Integrate, 1995, Vol.18A, n. 2, 108-129
school level: e;
mathematical subject: m; educational area: mr.
The role of analogical reasoning in learning mathematics is taken into account
in the general perspective of learning as a constructive process and a
continuous interaction between what is already known and what is to be
learnt. In education, analogical reasoning is commonly used to build new
patterns and solve new problems on the basis of the old ones. Analogical
reasoning can constitue a powerful didactic instrument, provided the student’s
ability in mastering the process of mapping. However, one should not neglect
that analogical reasoning could also induce incorrect conclusions, when
emphasis is given to specific, partial aspects. In short, analogy is recognizable
as a double edged weapon: as a means to generate new knowledge and as a
potential source of misconceptions. In learning mathematics, the ability to
perceive similarities and analogies plays a crucial role in mathematical
reasoning, problem solving and concept formation. Here a theoretical model
based on the Frege’s semiotic triangle and originally sketched to describe the
very nature of algebraic thinking (Arzarello, Bazzini, Chiappini, 1994) is used
to reconsider the underlying dinamics of analogical reasoning. Questions
related to analogical reasoning and learning mathematics are also treated in:
BAZZINI L.: 1994, 'Il ruolo del pensiero analogico nella costruzione di
conoscenze numeriche' ('The role of analogical thinking in the construction of
numerical knowledge'), in Numeri e proprietà, Atti del I Internuclei scuola
dell'Obbligo, Università degli Studi di Parma, (107-112).
115
BAZZINI L.: 1994, 'Il ruolo dell'analogia nell'apprendimento della matematica'
('The role of analogy in mathematics learning'). In Gallo E., Giacardi L.,
Pastrone F. (eds.) Conferenze e Seminari 1993-1994, Ass. Subalpina
Mathesis, Seminario di Storia delle Matematiche "T. Viola", Università di
Torino, (231-241).
BAZZINI L.: 1997, 'Revisiting analogy in learning mathematics', in
Mathematics Education and Applications, Proceedings of the First
Mediterranean Conference on Mathematics, Cyprus, 2-5 Jan. 1997 (174-181).
BAZZINI L. (ed)
La Didattica dell’Algebra nella Scuola Secondaria Superiore, Atti del V
Convegno Internuclei per la Scuola Secondaria Superiore, Pavia, 16-18 marzo
1995, ISDAF, PAVIA
school level: s;
mathematical subject: m;
educational area: cr, tt, e.
This books is a collection of papers presented at the Fifth Conference for the
Didactic Research Groups involved in secondary education (Pavia, 1995).
Here the list of contributors:
Accascina G. et al.: La preparazione degli studenti in algebra alla fine delle
scuole secondarie superiori.
Barbi G. et al. Un'indagine sulle difficoltà relative ai concetti di potenza,
funzione esponenziale e logaritmo.
Barbieri E.. et al.: Una strategia per il recupero.
Bazzini L.: Equazioni e disequazioni: riflessioni sul concetto di equivalenza.
Bovio M. et al.: Equazioni di primo grado nel biennio delle superiori
Cacciabue R.A.: Perché le matrici.
Cannizzaro L., Celentano A.: Rapporti fra aritmetica e algebra simbolica: un
test e una proposta di intervento nella scuola secondaria superiore.
Capelli L. et al.:La didattica dell'algebra nella scuola secondaria superiore.
Furinghetti F.: Una lettura della letteratura su insegnamento/apprendimento
dell'algebra a livello di scuola secondaria superiore.
Gallo E.et al.: La manipolazione algebrica: aspetti concettuali e procedurali.
Impedovo M.: Che cosa è davvero importante del calcolo letterale?
Malara N.A. : Mutamenti e permanenze nell'insegnamento delle equazioni
algebriche. Da un'analisi di libri di testo di algebra editi a partire dal 1880.
Paola D.: Ricomincio da ... N.
Rognoni D.: Aspetti didattici del simbolismo algebrico.
BAZZINI L.
Equazioni e disequazioni: riflessioni sul concetto di equivalenza ('Equations
and inequalities: reflections on the concept of equivalence'), In Bazzini
116
L.(Ed.) La didattica dell’Algebra nella Scuola Secondaria Superiore, Atti del
V Convegno Internuclei per la Scuola Secondaria Superiore, (Pavia, marzo
1985,), 1997, ISDAF, Pavia, 44-53.
school level: b, t; mathematical subject: al; educational area: mr.
Several research studies have pointed out difficulties emerging when students
face equations and inequalities. There is evidence that students often consider
an algebraic expression only as a string of symbols disconnected from any
semantics. In analogy with the studies carried out by Linchevski and Sfard,
we have investigated the students conceptions about questions like: What
does it mean to solve an equation (inequality)?; Which are the permitted
transformations?; When are two equations (inequalities) considered as
equivalent?. The results of this study confirm the hypothesis that students’
responses on the equivalence of equations and inequalities highly depend on
the presence of formal transformations.
This typical students’ behavior is interpreted here by applying the theoretical
model for algebraic thinking outlined by Arzarello, Bazzini and Chiappini
(1994) and based on the Fregean distinction between sense and denotation of
a symbolic expression.
BAZZINI L.
Analysis of a classroom episode by means of a theoretical model, in Les liens
entre la pratique de la classe et la recherche en didactique des
mathématiques, Actes de la CIEAEM 50, (Neuchatel, 1998), 1999, 292-296)
school level: e;
mathematical subject: ar; educational area: mr.
This report is a a contribuition to the debate on the relationships between
classroom practice and theoretical research, by the analysis of a classroom
episode in the view of a theoretical framework. Such a framework was
previously outlined as a result of the observation of students’ behavior when
facing algebraic problem solving. The model was successively applied to the
field of arithmetics: here similar dynamics emerged. The analysis of a
classroom episode shows that during the course of classroom interaction
students actively construct different relationships between signs/symbols and
meanings.
As a consequence, it seems interesting to focus on classroom episodes
readable in term of theoretical investigation, which, in turn, has developed
from the classroom observation.
This spiral development between theory and practice hopefully provides
research in mathematics education of fruitful implications for teaching. An
extended version of this paper can be found in:
117
BAZZINI L., 1998, 'Significati in gioco in un curriculum per le elementari'
('Meanings involved in a primary school curriculum'), in Gallo E., Giacardi
L., Roero C.S.. (eds.) Conferenze e Seminari 1997-1998, Ass. Subalpina
Mathesis, Seminario di Storia delle Matematiche "T. Viola", Università di
Torino (68-76)
BAZZINI L.
'On the construction and interpretation of symbolic expressions in algebra', in
Proc. of CERME I (First Conference of the European research in Mathematics
Education), Osnabrueck, 1998, in press.
school level: b; mathematical subject: al; educational area: mr.
Recent research studies have pointed out the crucial role of constructing and
interpreting letters in algebra. Many difficulties emerge because of the
incapability to relate the algebraic code to the semantics of the natural
language. A teaching experiment, carried out with 16 year old students,
attending the second year of the Gymnasium (a humanities oriented High
School) is described here. This experiment was aimed at analysing the
cognitive behavior of the students when facing learning situations dealing
with a productive use of symbols and their understanding. A short synthesis
of this article can be found in:
Bazzini L.: 1999, 'From natural language to symbolic expression: students’
difficulties in the process of naming', Proceedings of the 23rd Conference of
the International Group for the Psychology of Mathematics Education, Vol.I,
(263)
BAZZINI L., FERRARI M.
Experience of methodological and curricular innovation in primary school, in
Malara N., Rico L. (eds.) Proc. First Italian-Spanish Research Symposium in
Mathematics Education, Modena, 1994, 35-42.
school level: e;
mathematical subject: m; educational area: cr.
This paper sketches some main features of an experience of methodological
and curricular innovation in primary school, which has been carried out by the
Nucleo di Ricerca Didattica of the University of Pavia since 1985. Such
experience is located in the general situation of innovation which
accompanied the publishing of new Government Programs for primary
schools and characterized new trends in Mathematics Education. Firstly the
underlying philosophy of the project (cultural and methodological choices
118
and related implications for teaching) is tackled; secondly an example of
teaching activity is given. Finally, some considerations are sketched on the
basis of the work which has been done in the close cooperation with teachers.
The cultural choices of the project are also outlined in
Bazzini L., 1994, 'Cultural choices and teaching implications in primary
mathematics education', in Bazzini L., Steiner H.G. (Eds.) Proceedings of the
Second Italian-German Bilateral Symposium on Didactics of Mathematics,
IDM, Bielefeld (17-30).
The main choices are oriented to provide children opportunities of facing
mathematics in its double nature since the very beginning; i.e. as a strong
instrument to know and to interpret reality and as an exiting activity of the
human mind.
BAZZINI L., STEINER H.G. (eds.)
Proceedings of the Second Italian-German Bilateral Symposium on Didactics
of Mathematics, IDM, Bielefeld, 1994
school level: -;
mathematical subject: m;
educational area: cr, tt, e.
This book collects the contributions presented in the Second Italian-German
Bilateral Symposium on Didactics of Mathematics, held in Osnabrueck, April
1992.
Here a brief summary of the contents:
Tendencies and problems of curricular and educational innovations in primary
school mathematics (contributions by M.Bartolini Bussi, L.Bazzini, D.
Boenig, P.Boero, C. Morini, S. Schuette, J. Voigt, E.C. Wittmann)
The cultural and historical dimension of mathematics and their relation to
mathematics education (contributions by L. Cannizzaro, H.N. Janke,
M.Menghini, P.Schreiber, F. Speranza, H.G. Steiner)
Interaction between computer and mathematics education (contributions by E.
Cohors-Fresenborg, M.Fasano, F. Furinghetti, R. Hoelzl, C. Pellegrino, G.
Schrage, H. Schumann)
The mathematics classroom as a social system (contributions by E. Gallo,
N.A. Malara, A. Pesci, F. Seeger, H. Steinbring).
BAZZINI L., COLOMBI E., ZAMPIERI VENDER L.
Analisi del comportamento di bambini con difficoltà di apprendimento in una
situazione contestualizzata nel filone "Tempo" ('Analysis of low- attainer's
behavior in a unit included in the project 'Time''), L'Insegnamento della
Matematica e delle Scienze Integrate , 1996, Vol. 19A, N.1, (7-27).
school level: e;
mathematical subject: ar; educational area: m.
119
In the general framework of learning difficulties, this study concerns a
teaching experiment aimed at fostering children understanding of the time
sequence during a day.
The focus is on low-attainers, as observed during the experiment. The
authors’ hypothesis consists in proving that low-attainers performance
improves when instruction is grounded in strongly situated and socially
shared activities. There is evidence of the necessity of a global analysis of
child experiences, together with a proper intellectual reflection, in order to
help low-attainers to overcome their blocks in problem solving. Further
investigation of the cognitive behavior of subjects who are highly affected by
psychological malfunction can be found in:
BALDI P.L., BAZZINI L: 1998, 'La competenza numerica in soggetti con gravi
disturbi affettivi' ('Numerical competence in subjects with severe affective
difficulties'), in Matematica e affettività. Chi ha paura della matematica? (a
cura di I. Aschieri, M. Pertichino, P. Sandri, P.Vighi), Pitagora Editrice,
Bologna, (95-100).
Here the authors’ main purpose is the analysis of the basic knowledge used by
such subjects in numeracy. The counting procedure, accompanied by touching
and pointing, seems to be the key strategy which leads the subject to a correct
response.
BAZZINI L., FERRARI M., PESCI A., REGGIANI M.
Il progetto ‘Matematica come scoperta’: lo spirito continua (The project
'Mathematics as discovery': the spirit continues), L’Insegnamento della
Matematica e delle Scienze Integrate, 1995, vol. 18A-B n. 5, 1996, 445-473.
school level: b, t; mathematical subject: m; educational area: tt.
The project ‘Matematica come scoperta’ is considered the main work by G.
Prodi in the field of Mathematics Education. Such a project was conceived as
comprehensive of the whole mathematical curriculum for students attending
scientifically oriented secondary school.
This article deals with the Project as far as its structure, guidelines and
underlying philosophy are concerned. The authors, who cooperated with G.
Prodi in planning and experimenting the project, point out the main outcome
of the project itself: its seminal impact in curriculun innovation, as witnessed
by the successive trend of the Government Programs.
BECCHERE M., GRUGNETTI L., TAZZIOLI R., USELLI E.
120
'Sens commun et concept de volume' in C. Keitel et al. (eds) Mathematics
(education) and common sense, the challenge of social change and
technological development, Proc.CIEAEM 47 (Berlin,1995), 1996, 309-315.
school level: e, m, b, t; mathematical subject: g; educational area: pcb.
This paper deals with the concept of volume and common sense. Its purpose
is to contribute to the understanding of difficulties and misunderstandings in
the concept of volume coming from a sort of common sense and teaching
practices. The research, starting from Freudenthal's studies on the subject,
foresaw a 'vertical' investigation, which reached not only pupils of different
school sectors, but also adults and is concerned also with the analysis of the
definitions of 'volume' in different texbooks. The answers to a test of the
'vertical' investigation are analysed and compared with textbooks' definitions
of volume.
BERNARDI C.
I matematici e l’indirizzo didattico (Mathematicians and the educationoriented curriculum of the degree in mathematics), L’Educazione Matematica,
1995, XVI, serie IV, vol.2, 33-49
school level: u; mathematical subject: m; educational area: tt, i.
In the paper, mainly devoted to university teachers, various aspects of the
Italian degree in Mathematics are discussed related to the issue of teacher
formation. The lack of continuity between high school and university is
pointed out. Various issues related to the courses provided for Mathematics
freshman students are dealt with (like, for example, the relationships between
symbolic manipulation and spatial interpretation) and the main features of the
education-oriented curriculum (like degree thesis, contents of the courses,
other activities, ...) are analysed.
It is remarked that the social image of mathematics is affected by educational
practices more than by research or popularization. In the conclusion the
author urges that teachers should induce even people who will not be
involved in mathematical research (i.e. the majoirity of population) to enjoy
the ‘pleasure of doing mathematics’.
BERNARDI C
Riflessioni sull’uso del linguaggio in matematica (Reflections on the use of
language in mathematics), in I fondamenti della Matematica per la sua
didattica, 1997, Atti del Congresso Mathesis 1996, Verona, 81-89
school level: t, u; mathematical subject: m, l;
educational area: mr.
121
In this paper it is remarked that the logico-mathematical language is both a
tool and an object of investigation, a number of examples of common
linguistic use in mathematical contexts are given. In particular the differences
between mathematical and natural language (like the different role of
definitions) are examined; equivalent forms of ‘if-then’ constructions are
presented which seem to have a different meaning from the original statement
and some incorrect or improper mathematical statements are listed. It is also
remarked that the use of ‘e’ (‘and’ in english) and ‘o’ (‘or’ in english) in
mathematics is rather ambiguous (like for example in the solution of
inequalities or systems) and an explanation is proposed.
BERNARDI C.
How formal should a proof be in teaching mathematics?, Bulletin of the
Belgian mathematical Society, 1998, suppl.vol.5, n.5, 7-18.
school level: t;
mathematical subject: l; educational area: p.
The paper do not proposes to provide a thorough answer to the question in the
title but wants to underline the educational effectiveness of different styles of
proof. In particular the role of intuition in high school is discussed, taking into
account informal ‘proofs’ (including proofs with no words). On the other
hand the opportunity of presenting the logico-formal structure of some proofs
is enhanced. The paper contains a number of examples.
BERNARDI C.
Non abbiate paura (Don’t be afraid), in Matematica e affettività, Atti del VII
Convegno Nazionale Matematica e Difficoltà, 1998, Castel San Pietro, 1-8.
school level: m, b, t, u; mathematical subject: m, l; educational area: m.
Fear and anxiety have, in various contexts, both positive and negative
implications. Some negative aspects of fear in mathematics are discussed,
with reference to normal students, without taking into account specific
individual difficulties. The main claim is that often teachers unintentionally
transfer their own fears to students. After some general remark, some issues
are dealt with more closely which concern mathematical logic and some
subjets that for different reasons usually frighten students. At last, it is hinted
that the mastery of the links between syntax and semantics could be helpful to
overcome fear.
BOERO P.
122
Didactique des Théorèmes entre Mathématiques, Epistemologie et Sciences
Cognitives, Proc. CIEAEM 50, (Neuchatel 1998)1999, pp. 297-302.
school level: e, m; mathematical subject: m; educational area: p.
The aim of this contribution is to illustrate the research on the didactics of
theorems performed by the group of teachers and researchers in Genoa. The
emblematic character of this research is stressed as it concerns the plurality of
tools and methods (taken from different disciplines) which are needed if
research is to be aimed at producing results useful to interpret students'
difficulties and the preparation and analysis of innovations for classroom
work.
BOERO P.., GARUTI R.
Approaching Rational Geometry: from Physical Relationships to Conditional
Statements, Proc. PME 18, Lisboa, Portugal, 1994, vol. 2, 96-103
school level: m
mathematical subject: g
educational area: p
Reflections on some historical and epistemological aspects of the statements
of theorems in geometry suggested a teaching experiment with students in
grade VII, concerning the production of geometry statements and the
comparison between the statements produced and the statements contained in
the text-books. An analysis of the students' papers proves that through such
activities, in an adequate educational context, they are able to approach
geometry statements constructively.
BOERO P., SZENDREI, J.
Research and Results in Mathematics Education, in J. Kilpatrick & A.
Sierpinska (eds.), Mathematics Education as a Research Domain, 1997,
Kluwer Ac. Pub., pp. 197-212
school level: all; mathematical subject: m; educational area: tr.
We will propose a classification of scientific results in mathematics education
suitable (in our opinion) for analysing some of the present internal and
external difficulties and contradictions in the field of mathematics education.
Some contradictions connected with the requests of mathematicians and
mathematics teachers, school administrators, etc. will be discussed. Also
contradictions inherent to the effort of establishing mathematics education as
a specific field of research of the preparation of mathematics teachers, in
123
relation to present reality and current mathematicians' ideas about that
preparation.
BOERO P., CARLUCCI A., G.CHIAPPINI, FERRERO E., LEMUT E.
Pupils' cognitive development through technological experiences mediated by
the teacher, in: Wright J., Benzie D. (Eds.) Exploring a New Partnership:
Children, Teachers and Technology, Elsevier Science Publisher B.V., (NorthHolland), IFIP WG3.5, 1994, 103-120
school level: e, m; mathematical subject: m; educational area: cm cr, m.
In this paper we will refer to a long term study we are developing about the
classroom exploitation of some potentials of technological processes
concerning pupils' cognitive development. We will consider the opportunities
offered to the teacher by technological processes in order to develop logiclinguistic skills and problem solving strategies as well as production and
management of interpretative and planning hypotheses at large. In this
direction we also consider some innovative educational strategies suitable for
better exploiting the potentials inherent in technological processes.
BOERO P., CHIAPPINI G., GARUTI R., SIBILLA, A.
Towards Statements and Proofs in Elementary Arithmetic: An Exploratory
Study About the Role of Teachers and the Behaviour of Students, Proc. PME19, Recife, Brasil, 1995, vol. 3, 129-136
school level: m; mathematical subject: ar; educational area: p.
This report deals with the analysis of the behaviour of grade VI/VII students
whilst constructively approaching, in a suitable educational context,
statements and proofs of elementary arithmetic theorems. In particular, the
report deals in depth with the issues of the teacher as a mediator of the most
relevant characteristics of statements and proofs and the transition from the
statements produced by the students to the relative proofs.
BOERO P., CHIAPPINI G., PEDEMONTE B., ROBOTTI E.
The voices and echoes game and the interiorization of crucial aspects of
theoretical knowledge in a vygotskian perspective: ongoing research, Proc.
PME 22, Stellenbosch, South Africa, 1998, vol. 2, 120-127
school level: m, u; mathematical subject: m, hs; educational area: e.
124
This report presents some new findings about the "voices and echoes
game"(VEG), an innovative educational methodology conceived in a
Vygotskian perspective and aimed at approaching theoretical knowledge,
overcoming the intrinsic limitations of both traditional and constructivistic
approaches. Based on some improvements in the theoretical framework of the
VEG, new teaching experiments were performed. Analysis of student
behaviour allowed investigation of some individual and social cognitive
processes underlying the VEG, especially concerning the interiorization of
some aspects of theoretical knowledge.
BOERO P., DAPUETO C., FERRARI P., FERRERO E., GARUTI R., LEMUT E.,
PARENTI L., SCALI E.
Aspects of the Mathematics-Culture Relationship in Mathematics TeachingLearning in Compulsory School, Proc. PME 19, Recife, Brasil, 1995, vol. 1,
151-166
school level: e, m, b, t; mathematical subject: m; educational area: ap, e.
The purpose of this Research Forum presentation is to investigate some
cognitive
and didactic issues regarding the relationship between
"mathematics" and "culture" in teaching - learning
mathematics in
compulsory school. Our attention will focus, firstly, on how everyday culture
may be used within school to build up mathematical concepts and skills;
secondly, on the contribution that mathematics, as taught at school, may give
to everyday culture to allow (and spread) a "scientific" interpretation of
natural and social phenomena and, thirdly, on teaching mathematics as a part
of the scientific culture which ought to be handed over to the new
generations.
We will try to help make clear some potentials and some intrinsic limits of
teaching mathematics in "contexts", pointing out the role the teacher has to
play to make the best of such potentials and overcome such limits.
BOERO P., DAPUETO C., PARENTI L.
Research in Mathematics Education and Teacher Training, in Bishop A. (ed),
International Handbook of Mathematics Education, 1996, Kluwer Ac. Pub.,
pp. 1097-1122.
school level: e, m, b, t; mathematical subject: m; educational area: tr, tt.
The relationships between Research in Mathematics Education (R.M.E.) and
Mathematics Teachers Education (M.T.E.) may be considered under different
points of view, according to the ideas people have of both R.M.E. and M.T.E.
In this chapter we try to give a general outline of the problems. First of all, we
125
focus on some current ideas of M.T.E., trying to point out some of their
historical and present motivations. Then, we discuss what kind of tools and
results which R.M.E. offers today may be introduced into M.T.E., comparing
present needs with actual offers and pointing out some possible directions in
order to improve the present situation. Finally, we try to explain some specific
methodological issues, concerning the introduction of R.M.E. results and
tools into M.T.E., relatively indipendent of the choice of a peculiar
orientation in the field of R.M.E.
BOERO P., GARUTI R., LEMUT E.; GAZZOLO T., LLADO' C.
Some Aspects of the Construction of the Geometrical Conception of the
Phenomenon of Sunshadows, Proc. PME 19, Recife, Brasil, 1995, vol. 3, 310
school level: e;
mathematical subject: g; educational area: mr, pcb.
The persistence of "naive" conceptions relative to many natural phenomena in
subjects that have been learnt in school, a "scientific" interpretation for them,
and their difficulty in using school-learnt mathematical models to interpret
non-trivial situations raise interesting issues for psychological and educational
research. This report analyses some aspects relative to the passage to a
geometrical conception of the phenomenon of the Sun's shadows from the
"naive" non-geometrical conceptions that most 9/11 year-old students appear
to have of this phenomenon.
BOERO P., GARUTI R., LEMUT E. MARIOTTI M.A.
Challenging the traditional school approach to theorems: a hypothesis about
the cognitive unity of theorems, Proc. PME 20, 1996, Valencia, Spain, vol. 2,
113-120.
school level: m; mathematical subject: g; educational area: p.
The purpose of this report is that of highlighting the possibility that in an
adequate educational context the majority of grade VIII students successfully
implement a process of theorem (conjecture and proof) production,
characterised by a strong cognitive link between conjecture production and
proof construction. A detailed description is given of this process and of how
it surfaced in a teaching experiment organized by us. The conditions that may
have allowed the extensive implementation of the process in the classroom
are discussed and some educational implications are sketched.
BOERO P., GARUTI R., LEMUT E.
126
About the Generation of Conditionality of Statements and its Links with
Proving, Proc. PME 23, Haifa, Israel, 1999, vol. 2, 137-144
school level: e, m, u;
mathematical subject: m; educational area: p
Conditionality of statements (i.e. the fact that statements of most theorems are
implicitly or explicitly shaped according to the "if A then B" clause) has been
a peculiarity of theorems throughout the history of mathematics. The aim of
the research partially reported in this paper is to detect and describe a set of
processes of generation of conditionality in statements (PGC) that is wide
enough to cover the majority of PGCs that occur in different fields of
mathematics. In this paper we will describe four kinds of PGCs, along with
some productive links between these PGCs and the processes of construction
of proof.
BOERO P., GARUTI R., MARIOTTI M.A.
'Some dynamic mental processes underlying producing and proving
conjectures', Proc. PME 20, Valencia, Spain, 1996, vol. 2, 121-128
school level: m; mathematical subject: g; educational area: p.
The purpose of this report is the introduction and justification, on the basis of
a teaching experiment, of a hypothesis concerning the crucial role that can be
played by the dynamic exploration of the problem situation in the production
and proof of the conjecture required to solve the problem.. We will show how
students can generate the conditionality of the statement and the functional
connection with the subsequent proof through the dynamic exploration of the
problem situation.
BOERO P., PEDEMONTE B., ROBOTTI E.
Approaching Theoretical Knowledge through Voices and Echoes: a
Vygotskian Perspective, Proc. PME 21, Lahti, Finland, 1997, vol. 2, pp. 8188
school level: m; mathematical subject: m, hs; educational area: e.
This report deals with the ongoing construction of an innovative theoretical
framework designed to organise and analyse early student approach to
theoretical knowledge in compulsory education, the aim being to overcome
the limits of traditional learning and constructivist hypothesis. Referring to
Vygoskian analysis of the distinction between everyday and scientific
concepts and the Bachtinian construct of 'voice', and drawing on previous
teaching experiments (performed in Grade VIII), we hypothesise that the
introduction in the classroom of 'voices' from the history of mathematics and
127
science might (by means of suitable tasks) develop into a 'voices and echoes
game' suitable for the mediation of some important elements of theoretical
knowledge.
BONOTTO C.
Sull’integrazione delle strutture numeriche nella scuola dell’obbligo,
L’Insegnamento della Matematica e delle Scienze Integrate, 1995, 18A, n.4,
311-338.
school level: e, m; mathematical subject: ar; educational area: d.
In this paper we present the results of an investigation which has as its goal
the verification of knowledge regarding decimal and rational numbers in
children 10-14 years old. In particular we discuss how the pupils of the
compulsory schools are capable of receiving and assimilating the extension of
the number system from the natural numbers to that of decimals and fractions,
and later, integrating this extension into a single and coherent numerical
structure, in particular with regard to problems of ordering. This integration is
not a trivial extension, but is a very complex process, and indeed some
children might finish compulsory school before they comprehend it fully,
according to classical researchs (Nesher & Peled, 1986, Resnick, et al., 1989).
BONOTTO C.
Sul modo di affrontare i numeri decimali nella scuola dell’obbligo,
L’Insegnamento della Matematica e delle Scienze Integrate, 1996, 19A,
n.2,107-132.
school level: e, m; mathematical subject: ar; educational area: tcb.
In this paper we present the results of two questionnaires with the aim of
collecting information from elementary and middle school teachers regarding
the way in which they formulate the topic of decimal numbers in class. The
goal of this research was to find posssible connections between teaching
methods and difficulties in working with decimals wich has been observed in
earlier investigation (see studies of Hiebert, 1986, Nesher, 1986, and Even &
Tirosh, 1995). The results of this investigation has revealed that little time
and little energy are spent on building up meanings (all the attention is
focused on formal written rules and conventional aspects). In addition, the
little usually done to build up meanings is too partial towards fractions seen as
‘operators’and connections between decimals and decimal measurements
More generally, work done at school seems to have no connection with
everyday knowledge in the arithmetic field, that is to say there is no
128
connection with the rich experience students attain about numbers out of and
before primary school.
BOSCO A., DAPUETO C., GAGGERO M.T., MORTOLA C., TIRAGALLO G.
L'insegnamento della geometria nella scuola secondaria superiore - I e II parte
('The teaching of geometry in high school - I and II part), L’Insegnamento
della Matematica e delle Scienze Integrate, 1995, vol. 18 B, n.2-3, 135-146,
237-264
school level: b, t; mathematical subject: g; educational area: e, p, tt, cr.
The first part of this paper proposes some technical, didactical and
epistemological problems related to geometry and its teaching: • the teachers'
and pupils' difficulties, • the nature and role of mathematical proofs, • the
comparison among different (axiomatic and non-axiomatic) presentations, •
the comparison between "scholastic" and "out-of-school" geometrical
reasoning, the analysis of secondary school programs and textbooks. The
second part discusses the questions and develops them with historical and
foundational considerations. Some cultural and didactical aspects which are
basic for planning curricula and activities on geometry are pointed out and
clarified, and some didactical solutions are suggested. In particular, the
choices made in building the MaCoSa project for upper secondary school are
illustrated.
BOTTINO R.M., CHIAPPINI G.
ARI-LAB: models issues and strategies in the design of a multiple-tools
problem solving environment, Instructional Science, Vol. 23, n°1-3, 1995,
Kluwer Academic Publishers, 7-23.
school level: e;
mathematical subject: ar; educational area: cm.
In this paper we refer to a project aimed at designing, implementing and
evaluating a multiple-tools system to assist pupils in solving arithmetic
problems at the ages of 7-12.
The theoretical framework and the a priori analysis that have inspired the
design of the system are reported with regard to both cognitive and
educational aspects. A description of the main features of the system is
provided together with some findings from the first classroom tests of the
system.
The software engineering and ergonomic choices are justified by the analysis
of the cognitive processes and educational questions involved in the task of
building knowledge in arithmetic problem solving. From the software
engineering point of view, the system combines hypermedia and network
129
communication technologies with knowledge based systems; from the
cognitive point of view learning of the specific subject matter is the result of a
synergy of interpretation, communication and action processes that are
developed thanks to the mediation of the technology involved.
BOTTINO R.M., CHIAPPINI G.
User action and social interaction mediated by direct manipulation interfaces,
Education and Information Technology, IFIP TC-3 Official Journal, Kluwer
Academic Publishers, 3 (3/4), 1998, 203-216.
school level: e;
mathematical subject: ar;
educational area: cm.
In this paper we discuss a theoretical framework aimed to specify the
conditions under which the mediation offered by an educational system
(based on a direct manipulation interface) is effective for the teaching and
learning activity. We have worked out this framework on the basis of the
experience we developed in the design, implementation and experimentation
of systems for mathematics education.
BOTTINO R.M., FURINGHETTI F.
Teacher Training, Problems in Mathematics Teaching and the Use of
Software Tools, in D. Watson and D. Tinsley (Eds.): Integrating Information
Technology into Education, London: Chapman & Hall, 1995, 267-270.
school level: b, t; mathematical subject: m; educational area: tcb.
The teaching of mathematics is living a period of ferment and renewing. The
introduction of computer science plays the most relevant role in this renewal,
although other subjects as probability, statistics, logic are entered in school
practice. In order to investigate on how teachers interpret the renovation of
mathematics teaching we carried out a research on the present way of
teaching mathematics at the age 14-16 in our country which presents some
crucial problems which are both cultural and pedagogical. We focus in
particular on the problems of which mathematical abilities have to be pursued
at the age under discussion and of which content and methodology are more
suitable to achieve them.
We analyse the way in which teachers face these problems in school practice
since it is significant to outline the general methodology of their work in
classroom. The aim is to give a picture of the context in which the present
numerous proposals of curriculum reforms are introduced.
BOTTINO R. M., FURINGHETTI F.
130
‘The Emergence of Teachers’ Conceptions of New Subjects Inserted in
Mathematics Programs: the Case of Informatics’, Educational Studies in
Mathematics, 1996, vol. 30, 109-134.
school level: b, t
mathematical subject: cm
educational area: tcb
The changes in mathematical curricula induced by the introduction of
informatics in school represent the general framework of this research. In
particular we focus on the teacher’s role by analyzing the different choices
taken by mathematics teachers when faced with a curriculum reform induced
by the introduction of informatics in secondary school courses (age 14-16).
Our hypothesis is that these choices are the consequence of conceptions
teachers have about informatics and its teaching in relation to the teaching of
mathematics. Thus, through a case study research method, we focus on
mathematics teachers’ conceptions of informatics and its teaching. An attempt
is made at outlining a typology of these conceptions, based on the different
orientations identified.
BOTTINO R. M., FURINGHETTI F.
The Computer In Mathematics Teaching: Scenes From The Classroom,
Information and Communications Technologies in School Mathematics, D.
Tinsley and D.C. Johnson (eds.), London: Chapman & Hall, IFIP Series.,
Chapter 16, 1998, 131-139.
school level: b; mathematical subject: c; educational area: tcb, tt.
In this paper we analyse, through a case-study approach, the role assigned by
mathematics teachers to the use of educational software. We consider cases in
which teachers autonomously chose and use the software. Our analysis is
carried out by the direct observation of classroom activities. The data
collected are analysed according to a number of issues we organize around
some main areas. These areas have been identified as crucial in studying how
the use of technology affects the way in which mathematics is taught and the
way in which teachers perceive their role in classroom interaction.
BOTTINO R.M., FURINGHETTI F.
Mathematics Teachers, New Technologies And Professional Development:
Opportunities And Problems, in N. Ellerton (editor): Mathematics Teachers
Development: International Perpsectives, AU: Meridian Press, 1999, Section
1, pp. 1-11.
school level: e;
mathematical subject: ar; educational area: cm.
131
We set ourselves the task of investigating the ways in which teachers think
and feel about employing the computer in their mathematics teaching, how
their interactions with the computer influenced and were influenced by their
pedagogical approach, and how they integrated the computer into their
classroom practice. We consider, in particular, teachers involved with the
secondary school level (students' age: 14-18) and we fix our attention on the
use of educational software and software packages. We are interested in what
teachers actually do with software tools and why, as well as if the use of
technology have changed their teaching and their pedagogical approach.
Our research is grounded on our experience with a group of secondary school
mathematics teachers we have worked with since many years in projects
concerned with educational innovations in the classroom.
BOTTINO R.M., FURINGHETTI F.
Teaching Mathematics and Using Computers: Links between Teachers'
Beliefs in Two Different Domains, Proc. PME 18, Lisbona, Portugal, 1994,
vol. II, 112-119.
school level: b, t; mathematical subject: m; educational area: tcb.
The general framework of this research is the problem of the curricular
changes determined by the introduction of computers in school. We
investigate, through a case study methodology, how in-service mathematics
teachers are reacting to this introduction. In the paper we briefly outline the
methodology of our work and the context in which it is set. Then we identify
a number of issues we consider significant to investigate links between
teachers' beliefs in mathematics and the use of computers. Findings resulting
from the analysis of the case studies are presented according to these issues.
They allow to enlighten the links between beliefs in maths teaching and the
use of computers.
BOTTINO R. M., FURINGHETTI F.
The Computer in Mathematics Teaching: Scenes from the Classroom, in J. D.
Tinsley & D. C. Johnson (editors), Information and communication
technologies in school mathematics (IFIP TC3 / WG3.1), 1998, Chapman &
Hall, London, 131-139.
school level: b, t; mathematical subject: cm;
educational area: tcb.
This paper analyses, through examples, the role that computers could have in
concrete teaching activities. We consider, in particular, the use of software
(both educational and applicative) for mathematics teaching at upper
132
secondary school level (students aged from 14 to 18). At present, more and
more curricula foreseeing the use of software tools into school subjects are
designed. This implies that computer literacy is becoming less a subject in
itself and more a practical understanding of the capabilities and limitations of
computers in order to improve teaching-learning processes. The general
orientation seems to be the restructuring of education using technology by
developing: a) educational systems based on discovery and processes
construction in which students are actively involved; and b) carefully
designed educational settings and itineraries which integrate the use of
software systems. The methodology used in the research is direct observation
of the teachers behavior in classroom, in regular mathematical activities in
which computer is used.
BOTTINO R.M., FURINGHETTI F.
Teachers’ behaviours in teaching with computers, Proc PME 20, Valencia,
Spain, 1996, vol.2, 129-136.
school level: b; mathematical subject: c, g, al; educational area: tcb.
In this work we investigate, by means of interviews carried out with a sample
of upper secondary school mathematics teachers, the behaviours of teachers
when using educational software tools in their classrooms. In particular, we
examine the different choices teachers have autonomously developed at this
regard. We briefly outline the methodology of our work and we identify a
number of issues we consider significant to investigate how teachers'
behaviours in the use of computers influenced and was influenced by teachers'
general behaviour and beliefs in teaching mathematics. Some findings
resulting from the analysis of the interviews are presented according to the
identified issues.
BOTTINO R.M., CHIAPPINI G., FERRARI P.L.
Arithmetic Microworls in a Hypermedia System for Problem Solving, in L.
Burton and B. Jawrosky (eds.), Technology in Mathematics Teching: A
Bridge between Teaching and Learning, Bromley (U.K.), Chartwell-Bratt
Publishers, 1995, pp. 449-468.
school level: e;
mathematical subject: ar; educational area: cm.
In this paper we refer to the ARI-LAB system, an educational computer-based
system which combines hypermedia and communication technologies in order
to allow the user to build her/his own solution to a given arithmetic problem
133
by navigating through different integrated environments. In particular, we
focus on the arithmetic microworld which are included in the system The
discussion takes into account both pedagogical and technical aspects.
BOTTINO R.M., CHIAPPINI G., FERRARI P.L.
A hypermedia system for interactive problem solving in arithmetic, Journal of
Educational Multimedia and Hypermedia, AACE, Vol. 3, n° 3/4, 1994, pp.
307-326.
school level: e;
mathematical subject: ar; educational area: c.
This paper describes the first implementation of ARI-LAB, a system that
combines hypermedia and network communication technologies in order to
assist pupils in arithmetic problem solving.
The theoretical framework and the a priori analysis that have inspired the
design of the system are reported with regard both to cognitive and
educational aspects. The main features of ARI-LAB are presented and the
innovative aspects that in our opinion characterise it are pointed out. Software
engineering and user-interface choices are justified with particular reference
to hypermedia and human-computer interaction research. Moreover, the first
evaluation of the system, which was carried out with four deaf children in a
primary school, is presented and some findings are discussed.
BOTTINO R. M., CUTUGNO P., FURINGHETTI F.
Progettazione e utilizzo di un sistema ipermediale per la storia della
matematica (Planning and using a hypermedia for the history of
mathematics), L’Insegnamento della Matematica e delle Scienze Integrate,
1997, vol. 20A-B, 839-854.
school level: u; mathematical subject: hs;
educational area: tt.
This paper concerns a hypermedia (IPER-3) that we have planned to treat the
three ‘famous problems’ (squaring the circle, the duplication of the cube,
trisection of the angle). The aim of our work is to study the potentialities
offered by this kind of technology for presenting mathematical topics both in
teacher training courses and in classroom work.
The paper is organized as follows. In the first part we explain our choice of
history as knowledge field to work in and of the three ‘famous problems’. In
the second part we present the structure of IPER-3, focusing on the
technological choices that rely on interesting didactic issues.
In the third part we analyze the experience of use of this hypermedia that we
have carried out with university students. Eventually we outline some
possible developments for such a kind of activities.
134
BOTTINO R.M., CUTUGNO P., FURINGHETTI F.
Hypermedia as a means for learning and for thinking about learning, in T.
Ottmann & I. Tomek (eds.) Proc. ED-MEDIA/ED-TELECOM 98 10th World
Conferences on Educational Multimedia and Hypermedia and on Educational
Telecommunications”, AACE, USA: Charlottesville, Vol.1, pp. 144-149
school level: u; mathematical subject: hs;
educational area: tt.
The paper refers to a project aimed at designing, implementing and evaluating
a hypermedia system, IPER-3, facing the three ‘classical’ problems in the
history of mathematics (trisection of the angle, quadrature of the circle,
duplication of the cube). The aim of the project is to study the opportunities
offered by this kind of technology to the presentation of mathematical topics
both in teacher training courses and in classroom work.
BOVIO M., REGGIANI M., VERCESI N.
Problemi didattici relativi alle equazioni di primo grado nel biennio delle
superiori (Didactic problems about the first grade equations in the first two
years of high school), L'Insegnamento della Matematica e delle Scienze
Integrate, 1995, vol.18B, n.1, pagg.7-32
school level: b; mathematical subject: al; educational area: cr.
The article is the result of the work of some of the members of the Didactic
Research Group of Pavia. Working on difficulties Upper Secondary School
students meet in solving equations, the authors sketch a didactic outline to
follow, connecting it to the work usually done in Junior Secondary School.
They focus on nodal points which are not always clear in textbooks and which
teachers not always make so clear as they should. The topics relevant to
equations which are dealt with, are the traditional ones, but the purpose is to
make them become basic points in a didactic planning and not only
instruments as they are usually considered.
CAPELLI L., DAPUETO C., GRECO S.
Software per l'insegnamento della matematica: rappresentazione grafica di
funzioni ed equazioni (Software for mathematics teaching: representing
graphs of functions and equations), L’Insegnamento della Matematica e delle
Scienze Integrate, 1999, vol. 22B, n. 2
school level: b, t; mathematical subject: m, c;
educational area: cm.
135
In this paper some questions related to making and using software for
graphing functions and equations are discussed, in particular about scaling
and choosing domains, representing sequences of points, discontinuous
functions and implicit equations, problems connected with internal
representation of numbers. These issues are discussed from both technical and
didactical points of view, and placed in the general context of using computer
in teaching/learning mathematics in upper secondary school. Free software
carried out (and used in an upper secondary school project) by MaCoSa
Group (http://www.dima.unige.it/macosa) is presented and its features and
problems are pointed out and compared with the ones of commercial software
(spreadsheets, Derive, Maple).
CAREDDA C.
Matematica e difficoltà (Mathematics and Difficulties), Notiziario UMI,
October 1998, 29-36
school level: e, m; mathematical subject: m; educational area: d, m.
This article, whilst not chronologically following the ministerial regulations
emphasising the need for education offering everyone equal opportunities for
learning, highlights the contribution of outline law no. 104 of 1992. The stateof-the-art of didactical research in the sector of mathematics and difficulty
conducted by the Inter-University Research Group on Mathematics and
Difficulty (GRIMED) is described.
CAREDDA C., PUXEDDU M.R.
Adattamento di unità didattiche sulla probabilità a diverse situazioni di
apprendimento (Adapting probability teaching units to different classroom
situations), Induzioni, 1995, n. 11 87-101
school level: c, e; mathematical subject: p; educational area: cr.
The importance of gradualness in educational proposals and the
multifunctional role of each proposal is underlined here. After a brief
overview of the theories on learning and the very meaning of learning, a
didactic unit forming part of an itinerary on the analysis of situations of
uncertainty is described. This is a game, proposed in some 2nd year
elementary school classes and adapted to different learning situations, the aim
of which, at the pedagogical level, is to stimulate an analytical attitude, firstly
qualitatively and secondly quantitatively, to deduce estimates of probability.
CASELLA F., CIMADOMO M.R., DE LUCA G., FASANO M., GRANDE R.
136
Concepts in network. From the Conceptual Map execution to an hypermedia
production, Masson, Milano, 1998
school level: e;
mathematical subject: m; educational area: cm, tt.
This text is based on the results of some research carried out by the authors in
the last ten years in the field both of in-service teacher training and of
application of multimedia computer technologies to didactics. Some factors
are particularly emphasized as they influence the teacher behaviour and
attitudes in the light of technological innovation process in learning and
teaching: These factors are the multimedia environment which eases the
approach to computers, the development of conceptual maps which are
considered as an important instrument for the relational organization of
disciplinary and pluridisciplinary knowledge and the planning, seen as a basic
instrument in order to develop the capacity for resource utilization.
From these preliminary remarks, the didactic proposal, illustrated in this text,
is the realization of a multimedia product by the teacher, who is stimulated to
an epistemological reflection both on his/her own discipline and on the
numerous connections with other disciplines, as well as by software and their
lesson plans..
CASSANI A., D’AMORE B., DE LEONARDI C., GIROTTI G.
Problemi di routine e situazioni "insolite". Il "caso" del volume della
piramide, L’Insegnamento della Matematica e delle Scienze Integrate, 1996,
19B, 3, 249260.
English translation: Routine problems and “unusual” problems. The “case” of
the volume of a pyramid, in: A. Gagatsis and L. Rogers (eds.), Didactics and
History of Mathematics, Erasmus ICP 95 G 2011/11, Thessaloniki 1996, 7382.
Spanish translation: Problemas de rutina y situaciones “insolitas”. El “ caso”
del volumen de la piramide, Números (Tenerife, Canarias, Spain), 38, 1999,
21-32..
school level: m; mathematical subject: g; educational area: mr.
Main theoretical frame: In this paper we are examining the results of a
problem about an unusual situation which was presented to the students in
contrast to an analogous routine problem. The particular case was to calculate
the volume of a real pyramid. The task was assigned to students aged thirteen,
good at solving the related formal problem. The students’ responses and their
choice of strategy are emphasised and analysed. We made particular reference
to works by E. Fischbein, N. Fisher and H. Wertheimer.
137
Research problem: Some pupils that can be considered good solvers of formal
geometric problems cannot deal with the same real problems i.e. related to
real objects. This show the complete separation between real problems and
their mathematical formalization and that pupils give to formal problems
implicit meanings referred to school background and not to the real external
environment.
School-level: 3rd class (the last one) of lower middle school (pupils aged 1314 years).
CASTAGNOLA E., JOO C., PESCI A.
Adjusting the didactic itinerary to the pupils’ proposals: “Federico’s
Theorem” case, in Abrantes P., Porfirio J., Baia M. (Eds.),Proc. CIEAEM 49,
(Sétubal, Portugal,1997), 1998, 233-240.
school level: m; mathematical subject: ar; educational area: cr, d, pcb.
In the framework of the current perspectives of constructivism, the paper
describes an episode which occured during a didactical experience on the
construction of proportional reasoning with students aged 12-13. The
resolution strategy proposed by four students and explained in particular by
one of them, was different from the strategy foreseen by the teacher.
Nevertheless it was correct and it was the occasion for interesting discussions
in class.
The specific mathematical context is put in evidence and described in detail,
with particular attention to some significant student protocols. The aim of the
paper is that of underlining the fact that it is not easy to adjust the didactical
itinerary to students’ needs and curiosity. Their solution strategies and their
verbal explanations are often difficult to interpret and understand fully. But it
is essential that teachers learn to do this better, with the objective of a real
improvement in the quality of mathematics education.
CASTRO C., LOCATELLO S., MELONI G.
Il problema della gita. Uso dei dati impliciti nei problemi di matematica (‘The
trip problem.
Use of implicit data in mathematics problems’), La Matematica e la sua
didattica, 2, 1996, 166-184.
school level: e;
mathematical subject: m;
educational area: m, mr.
Main theoretical frame: This work aims to explain the difficulties met by 11
year-old children who attend the fifth year of the Italian primary school. It is
connected to research of the same type by Paolo Boero and Bruno D’Amore,
and in connexion with research of Guy Brousseau (didactic contract), Colette
138
Laborde (the use of everyday language in mathematicss), Alan H. Schoenfeld
(metacognition) and Gerard Vergnaud (concepts and schemes).
Research problem: In this paper we analyse the question of the splitting up of
a complex, mathematic problem into three components in order to understand
if a gradual approach to the problem could help children to imagine better the
contest of the situation. In the researche the children are asked to solve a
problem concerning a school trip. The return from the trip is not mentioned in
the problem (implicit datum). The results of this research underline the
importance of the “didactic contract”: on it can depend the success or the
failure of the child’s performance in solving problems with missing data.
CROSIA L., GRIGNANI T., MAGENES M. R., PESCI A.
La divisione tra polinomi: una proposta didattica per la scuola media
superiore (The polynomial division: a didactical proposal for senior
secondary students), L’Insegnamento della Matematica e delle Scienze
Integrate, 1996, vol. 19B, n. 1, 1996, 11-28.
school level: b; mathematical subject: al; educational area: cr.
The article describes a didactical experience with 15-16 year old students
which is centered on the construction of the algorithm of division between
polynomials in one variable.
From a theoretical point of view the analogy with the algorithm of repeated
differences in the frame of integer numbers is the idea which guides students’
research.
From a methodological point of view the discussion in small working-groups
and the collective discussion with teachers are the modalities which in our
opinion improve students’ participation and therefore their performance.The
objective of the proposal is not only that of making possible a better
understanding of the reason of the algorithm, but also that of showing to
students how it is possible to take part in the construction of mathematics
itself.
The article presents the texts of the three working-groups, the possible
answers to questions posed and, in detail, the results obtained in a second
class of the Industrial Technical Insitute “Cardano” in Pavia.
D’AMORE B.
Considerazioni su alcuni aspetti del comportamento logico e strategico degli
studenti al momento della risoluzione di problemi di matematica in àmbito
scolastico (‘Remarks on some aspect of logical and strategic behavior of
students when solving school mathematics problems’), L’Insegnamento della
Matematica e delle Scienze Integrate (Paderno, Italy), in press.
139
school level: e, m, b;
mathematical subject: m; educational area: mr.
Main theoretical frame and research problem: In this paper we explain the
behaviour and the logic actually used by students when they are searching for
strategies to solve school problems. Students protocols collected from
different investigations by the author over many years have been used to
arrive at a definition and exemplification of this idea. An analysis has been
made in order to distinguish and classify behaviours so that some causes of
verbal problems related to the texts themselves may be discovered.
School-level: primary school (pupils aged 6-11 years) and lower middle
school (pupils aged 11-14 years).
In the same field, see:
D’AMORE B., GIOVANNONI L. (1997). Coinvolgere gli allievi nella
costruzione del sapere matematico. Un’esperienza didattica nella scuola
media. La Matematica e la sua didattica (Bologna, Italy), 4, 360-399.
School-level: lower middle school (pupils aged 11-14 years).
D'AMORE B., MARTINI B. (1997). Contratto didattico, modelli mentali e
modelli intuitivi nella risoluzione di problemi scolastici standard. La
Matematica e la sua didattica (Bologna, Italy), 2, 150-175.
Spanish translation: Contrato didáctico, modelos mentales y modelos
intuitivos en la resolución de problemas escolares típicos, Números (Tenerife
- Canarias, Spain), 1997, 32, 26-32.
French translation: Contrat didactique, modèles mentaux et modèles intuitifs
dans la résolution de problèmes scolaires standard, Scientia Paedagogica
Experimentalis (Gent, Belgium), 1998, XXXV, 1, 95-118.
English translation: The Didactic Contract, Mental Models and Intuitive
Models in the Resolution of Standard Scholastic Problems, in: A. Gagatsis
(ed.), A multidimensional approach to learning in mathematics and sciences.
Intercollege Press, Nicosia, Cyprus 1999, 3-24.
School-level: primary school (pupils aged 6-11 years), lower middle school
(pupils aged 11-14 years) and high school (pupils aged 14-19 years).
D’AMORE B. (1999). Scolarizzazione del sapere e delle relazioni: effetti
sull’apprendimento della matematica. L’Insegnamento della Matematica e
delle Scienze Integrate (Paderno, Italy), 22A, 3, 247-276.
An ample summary in Spanish appears in: Resúmenes de la XIII Reunión
Latinoamericana de Matemática Educativa, Universidad Autonoma de Santo
Domingo, Santo Domingo, República Dominicana, 12-16 luglio 1999, 27.
Spanish translation: Relime (Mexico D.F., Mexico), in print.
School-level: primary school (pupils aged 6-11 years), lower middle school
(pupils aged 11-14 years) and high school (pupils aged 14-19 years).
D’AMORE B., SANDRI P.
140
Fa’ finta di essere.... Indagine sull'uso della lingua comune in contesto
matematico nella scuola media (Imagine you are .... An investigation on the
use of ordinary language in mathematical setting in middle school),
L’Insegnamento della Matematica e delle Scienze Integrate (Paderno, Italy),
19A, 3, 1996, 223-246.
Spanish translation: “Imagina que eres ...”. Indagación sobre el uso de la
lengua común en contexto matemático en la escuela media, Revista EMA
(Bogotà, Colombia), 1999, 4, 3, 1-26.
school level: m; mathematical subject: m; educational area: mr.
Main theoretical frame: This paper presents an investigation about the use of
spoken language in a mathematical context and the production of external
models of the student's deep ideas of some elementary concepts. We mainly
refer to studies about intuition by E. Fischbein and about communication of
the spoken language in Mathematics class by C. Laborde and by H. Maier.
Research problem: In our work we show that students really hardly use
natural language when they expose their ideas about mathematical themes;
nevertheless, by the use of decontextualization techniques, we do persuade
some pupils to express their own images in order to communicate to others: in
those cases we achieve sometimes illuminating and convincing evidence that
the use of everyday language can be a benefit in the activity of
communication in classroom as regard mathematical themes.
In the same field, see:
D'AMORE B., MARTINI B. (1998). Il “contesto naturale”. Influenza della
lingua naturale nelle risposte a test di matematica. L’Insegnamento della
Matematica e delle Scienze Integrate (Paderno, Italy), 21A, 3, 209-234.
Spanish translation: Suma (Sevilla, Spain), 30, 1999, 77-87.
School-level: last class of high school (pupils aged 18-19 years).
D’AMORE B., FRANCHINI D., GABELLINI G., MANCINI M., MASI F.,
MATTEUCCI A., PASCUCCI N., SANDRI P.
La ri-formulazione dei testi dei problemi scolastici standard, L’Insegnamento
della Matematica e delle Scienze Integrate, 1995, 18A, 2, 131-146.
English version: ‘The re-formulation of text of standard school problems’ in:
A. Gagatsis and L. Rogers (eds.), Didactics and History of Mathematics,
Erasmus ICP 954 G 2011/11, Thessaloniki 1996, 53-72.
school level: e, m; mathematical subject: m; educational area: mr
Main theoretical frame: Our work refers to the problems of rewriting
(implicitly or mentally) problem texts, and to the use of natural language in a
mathematical context. The difference in behaviour in such a situation between
141
beginners and experts is also considered with reference to school problem
solving (R. Glaser; H. Maier).
Research problem: We examined texts re-written by pupils in the classroom,
from the texts given by teachers, using individual or collective interview and
by discussions. We verified that, while the texts solved by adults were
discussed and re-written, the texts elaborated by pupils and proposed as a new
verbal problem to other classes produced few or no discussions. Nevertheless,
in spite of explicit statements by pupils, this situation does not lead to better
results with regard to resolutions, and this is rather interesting from the
metacognitive point of view.
School-level: primary school (pupils aged 6-11 years) and lower middle
school (pupils aged 11-14 years). In the same field, see: D’Amore B. (1996).
Schülersprache beim Lösen mathematischer Probleme. Journal für
Mathematik-Didaktik (Stuttgard, Germany), 17, 2, 81-97.
School-level: primary school (pupils aged 6-11 years), lower middle school
(pupils aged 11-14 years) and high school (pupils aged 14-19 years).
D'AMORE B
‘Oggetti relazionali e diversi registri rappresentativi: difficoltà cognitive ed
ostacoli’ (‘Relational objects and different representative registers: cognitive
difficulties and obstacles’) L'educazione matematica , 1, 1998, 7-28.
Spanish translation: Uno (Barcelona,), 1998, 15, 63-76.
school level: e, m, b, t; mathematical subject: m; educational area: mr.
Main theoretical frame: Our work is referred to problems of passages between
different representative registers, so particularly to works by R. Duval.
Research problem: We consider as “relational object” a message related to
binary relations which can be formulated in different linguistic registers and
we try, first of all, to point out if pupils can realize that such message is
univocal from a semantic point of view. We then consider which of these
messages is easier to undersatnd. We observed that, while the percentage of
acknowledgement of semantic univocalness is rather high, there are strong
dislikes by some students for some linguistic registers, in spite of what could
appear in classroom practice, where usually different registers are mixed as if
their acceptance is taken for granted and their choice neutral. From the
didactic point of view, we can suggest that making explicit the uses and the
passage from one register to another should be emphasised.
School-level: primary school (pupils aged 6-11 years), lower middle school
(pupils aged 11-14 years) and high school (pupils aged 14-19 years).
142
DA PONTE, J.P., BERGER, P., CANNIZZARO L., CONTRERAS L., SAFUANOV
I.
Research on Teachers' Beliefs: Empirical Work and Methodological
Challenges, in Krainer K. & Goffree F. (eds.), On Research in Mathematics
Teacher Education, Forschungsinstitut für Mathematikdidaktik, Osnabrück,
1999, 79-97.
school level: c, e, m, b, t, u;
mathematical subject: m; educational area: tt,
tcb
The chapter provides an overview of the empirical research regarding
teachers' beliefs towards mathematics and mathematics teaching and learning,
with a special emphasis on teachers' beliefs regarding selected mathematical
topics and problem solving. Some issue regarding the change of beliefs are
also discussed; the chapter ends with a brief analysis of methodological
questions implied in this field of research.
Our choice has been to focus on beliefs referred to single mathematical
concepts or to systems of concepts, meta-concepts, meta-aptitude or ability to
cope in the activity involving proof, recursion and problem solving,
mathematical-epistemological
roots,
mathematical-cognitive
roots,
mathematical-social aspects, and computer world views that are interrelated to
many of the fields previously mentioned.
DAPUETO C, PARENTI, L.
Contributions and Obstacles of Contexts in the Development of Mathematical
Knowledge, Educational Studies in Mathematics, 39, 1-21
school level: e, m, b, t; mathematical subject: m; educational area: e, mr, tr,
cr.
Over the past two decades, the failure of "new mathematics" has contributed,
together with other factors, to the development of a "movement", grounded in
theory and practice, which has focused renewed attention (in the planning of
mathematics curricula and in the study of concept formation) on the uses of
mathematics and its out-of-school applications. This paper proposes a
framework based on the epistemological concept of model and the didactical
one of field of experience in order to discuss the nature of the relationships
between contexts and the formation of mathematical knowledge and tackle
some educational, cultural and cognitive problems of situated teachinglearning: • forms of connection between mathematics and contexts, •
normative and descriptive aspects of mathematization, • risk of predominance
of mathematical point of view, • control over shifts between different layers
of meaning, and other problems. These issues are illustrated with some
143
examples based on the projects Genoa Groups have carried out at various
school levels.
DELL’AQUILA G., FERRARI M.
La lunga storia dei numeri interi relativi (The long history of relative
integers), Parte I, II, III, IV, V, L’Insegnamento della Matematica e delle
Scienze Integrate, vol. 17A n.2, 1994, 145-157, vol. 17A n. 3, 1994, 247-265,
vol. 17A n. 4, 1994, 335-361, vol. 18A n. 4, 339-363, vol. 19A n. 4, 1996,
311-339.
The first article deals with some epistemological obstacles to the knowledge
of integers and relates about the contribution given to the development of
integers’ theory by Babylonian and Chinese mathematicians and by
Diophantus from Alexandria.
school level: t, u; mathematical subject: hs;
educational area: tt.
The second part describes in detail the Hindu mathematicians’ contribution to
the birth and first development of relative integers. It also relates about the
Arabic mathematics approach and the first integers theory in the Christian
Medieval Europe.
The third paper is about mathematical achievements in Italy during XV and
XVI centuries towards relative integers theory. The contributions given by
distinguished authors such as Luca Pacioli, Gerolamo Cardano and Raffaele
Bombelli are chiefly analysed.
Part IV deals with mathematicians attitude towards relative integers during
XV and XVI centuries, in some european countries such as France, Germany,
Belgium and Great Britain. Scientists’ fear in accepting negative numbers and
negative roots in equations is here discussed.
Part V deals with the seventeenth century. It is in this period that scientific
correspondence became more and more usual and scientific Academies were
born in Italy, England and France. The mathematicians’ attitude towards
negative numbers is here studied and authors as Albert Girard, René
Descartes and John Wallis are considered in detail.
DELUCCHI S
Software per l'insegnamento della matematica: introduzione alla statistica
('Software for the teaching of mathematics: introduction to statistics'),
L'Insegnamento della Matematica e delle Scienze Integrate, 2000, vol. 23B/2,
177-190
school level: b; mathematical subject: s; educational area: cm.
144
This paper, through various examples of modelling activities, intends to point
out the role of computer in teaching-learning Statistics: • nowadays it is an
essential and integral tool in statistical activities, both in elaborating data and
in creating and developing new concepts, • it allows a natural combination of
reflective and experiential aspects and shifts the boundary between hard and
soft skills, so that its use in teaching permits to introduce otherwise too
difficult concepts and properties,• it helps and stimulates integration with
other mathematical fields. Free software carried out (and used in an upper
secondary
school
project)
by
MaCoSa
Group
(http://www.dima.unige.it/macosa) is presented and its features are pointed
out and justified with epistemological and didactical arguments.
DEMATTÈ, A.
Storia, pseudostoria, concezioni (History, pseudo-history, conceptions),
L’Insegnamento della Matematica e delle Scienze Integrate, 1994, vol.17B,
269-281.
school level: m; mathematical subject: hs;
educational area: pcb.
The author considers history of mathematics as an experiential field in which
students may acquire mathematical knowledge. In the paper he describes the
experiment carried out with students aged from 11 to 13. Students were asked
to express their conjectures on the birth of mathematical ideas and theories
and on the way of working of mathematicians. The work was partly
performed in little groups, partly individually. This activity was used by the
author to analyze the conception about the (supposed) historical development
of mathematics and about the nature of mathematics itself. From the work it
emerged a pseudo-history of mathematics and rooted myths about
mathematics. The author outlines also didactic implications.
DETTORI G., LEMUT E.
Relating effective Representations and Hypothesis Production in Arithmetic
Problem Solving, Proceedings CIEAEM 46, Toulose, 1994, 198-205
school level: m; mathematical subject: m; educational area: mr, tr.
Producing hypotheses is very important in problem solving in order to
understand the aim of the problem and the relations among the data. However,
a resolution process can also be viewed as a sequence of interpretations and
production of various external representations. We discuss the relations
between hypothesis and representation production in problem solving in
elementary school, viewing a resolution process as a sequence alternating
representations and hypotheses. These hypoteses and representations are
145
strongly connected with each other since each hypotesis can give rise to a
representation, and each representation, working as additional problem data,
can give rise to a new hypotesis which is influenced by the characteristics of
the employed representation system.
DETTORI G., LEMUT E.
External Representations in Arithmetic Problem Solving’ in Mason J.,
Sutherland R. (Eds.), Exploiting Mental Imagery with Computers in
Mathematic Education, NATO ASI Series F,
1995, Vol.138, Springer-Verlag, pp.20-33
school level: e;
mathematical subject: ar; educational area: mr, v.
We discuss the role of external representations in arithmetic problem solving
activities in elementary school (age 6-11). The analysis is made by referring
to a curricular project which emphasizes the relationship between achieving
arithmetic competences and solving problems. First we analyse the role and
the components of external representation in a pen-and-paper environment,
then we discuss different characteristics and impact of using representations
for arithmetic problem solving in a hypermedia environment.
DETTORI G., GARUTI R., LEMUT E., NETCHITAILOVA L.
An Analysis of the relationship between Spreadsheet and Algebra, in:
Technology in L.Burton and B.Jawrosky (eds.), Mathematics Education,
Chartwell-Bratt Publishers, 1995, 261-274
school level: m, b; mathematical subject: al;
educational area: cm, d, m.
In recent times, the spreadsheet has been suggested as a tool for teaching
algebra in intermediate high school. Our a-priori analysis of the relationship
between streadsheet and algebra shows the inadequacy of this tool to express
the fundamental characteristics of algebra, that is, the manipulation of
algebraic variables and relations, which make algebra suitable as a formalism
for describing models. However, with the attentive guidance of a teacher, the
spreadsheet can become a useful tool for motivating the introduction of some
concepts of algebra and for reflecting on different resolution models.
FASANO M.
Arithmetic and computer science in the primary mathematical education, in
Quaderno n. 23, Corso di formazione M.P.I.- UMI, Liceo Scientifico
“Vallisneri” - Lucca, 90-96.
school level: e;
mathematical subject: ar; educational area: cm.
146
In this text the theoretical and applicational aspects of computer science and
the influence they have on mathematics that is taught in Primary School are
emphasized. In particular, the activities based on both procedural and
relational thought are analysed and described. Furthermore, the development
of computer systems is presented, emphasizing the didactic importance of the
pocket-calculator, considered as an instrument for discovering and controlling
numerical regularities and algebraic properties.
FERRANDO E.
A Multidisciplinary Approach to the Interpretation of Some Difficulties in
Learning Mathematical Analysis, Proc. CIEAEM-50, (Neuchatel
Switzerland,1988), 1999, 308-312.
school level: u
mathematical subject: c
educational area: e
This paper focuses on the plurality of disciplines (such as philosophy, logic,
semiotics, psychology, epistemology and didactics) which provide the tools
required to interpret students' difficulties in the approach to Mathematical
Analysis, especially as concerns: (a) the interpretation of symbols of function
and (b) the learning of the concept of monotonic functions. The reported
study consisted of two subsequent phases: in the first phase tools of different
disciplines were used to develop an a-priori analysis about possible
difficulties concerning (a) and (b); in the second phase I have analyzed some
activities of a group of 25 Chemistry students that were following a Calculus
and Analitic Geometry course at the University of Eastern Piedmont at
Alessandria.
FERRARI M.
Continuità: utopia possibile (Continuity: a feasible utopia), L’insegnamento
della matematica
e delle scienze integrate, 1996, vol. 19A n. 3, 1996, 207-222.
school level: u; mathematical subject: m; educational area: tt.
In this paper the great problem of the cultural and educational continuity
through the passage from one school level to the next are widely discussed.
The author clears up the obstacles to the realization for such a continuity by
showing prerequisites and elements which currently assist it.
FERRARI M
“... Gli eredi riconoscenti divisero” (“... Heirs grateful shared”),
L’Insegnamento della Matematica e delle Scienze Integrate, 1997, vol. 20A-B
n. 6, 1997, 647-680.
147
school level: b, t; mathematical subject: ar, al;
educational area: tt.
In this article the author unfolds, sometimes jokingly, the concept of
divisibility. It is a concept with a long history; a fertile matter linked with
many other concepts; a pervasive concept because it originates from natural
numbers, it spreads to relative whole numbers, polynomials and homogeneous
quantities, reaching its height in Euclidean rings.
FERRARI P. L.
On some factors affecting advanced algebraic problem solving, in Gutierrez,
A. & L.Puig (eds.), Proc. PME 20, Valencia, Spain, 1996, vol.2, 345-352
school level: u; mathematical subject: al; educational area: mr.
This paper focuses on the learning of algebra at undergraduate level. Some
general hypotheses on cognitive, didactical and linguistic factors affecting
algebraic problem solving are discussed and tested. In particular, the duality
process/object is taken into account in order to explain students' problem
solving performances. Some empirical findings are presented concerning the
resolution of a sequence of algebra problems by a group of freshman
computer-science students over a four-months term. Students' results in the
whole sequence are compared to their results in other subject matters. Some
examples of correlation between linguistic skills and performance in algebra
are shown.
FERRARI P. L.
Action-based strategies in advanced algebraic problem solving, in Pehkonen,
E.
(ed.), Proc. PME 21, Lahti (Finland) 1997,.vol.2, 257-264
school level: u; mathematical subject: al; educational area: mr.
This paper examines the strategies adopted by a group of undergraduates in
order to solve a set of problems involving divisibility. The focus is on actionbased strategies, i.e. on strategies depending on physical manipulations which
are performed with little semantical control. It is shown that problems
requiring relational knowledge or impredicative reasoning may result in
difficulties for a number of students even if only elementary concepts and
methods are involved. Dubinsky’s Action - Process - Object - Scheme
framework has proved a useful tool to interpret students’ behaviors, even
though other aspects, related to the purposes of students’ efforts, are to be
taken into account.
148
FERRARI P. L.
The influence of language in advanced mathematical problem solving’, in
Proc. PME 22, Stellenbosch (South Africa), 1998, vol.4, 252.
school level: u; mathematical subject: m; educational area: mr.
The aim of this short paper is to study the relationships between some
obstacles in the learning of mathematics at tertiary level and students'
linguistic competence. Sometimes, in standard educational practice, the role
of languages is underrated, whereas, conversely, a poor mastery of language
may induce students to choose inadequate strategies. Some examples are
presented in order to point out some aspects of the influence of language (and
contract) on students' behaviors. In the second part of the paper some data on
the solution of an arithmetic problem by a group of freshman computer
science students are analyzed in a more detailed way, comparing the effects of
three different forms of presentation of the problem on students' strategies.
FIORI C., PELLEGRINO C.
Configurational theorems and coordinatization of the affine planes, and In
search of the lost affinities, La Matematica e la sua Didattica, 1995, n. 4,
431-445, and 1996, n. 1, 46-56
school level: u; mathematical subject: g; educational area: cr, e.
The non-Euclidean revolution has imposed the search for a foundation of
mathematics. Therefore the language of mathematics has become more
abstract and formal. As a consequence, in the last decades the university
teaching of mathematics has absorbed these features and eventually lost all
connections with the traditional language and approach.
This has gradually turned the culture and training of the new generations of
mathematicians much poorer and fragile. In order to contrast this fatal trend
that strongly conditions the quality of mathematics teaching in all school
levels, we started a study aimed at highlighting the links between the
traditional language of geometry and the current one, so as to allow us to
appreciate the advantages and qualities of both approaches.
Two papers have appeared from within this study. In the first, in order to
illustrate an important theorem of the foundation of Geometry, we face the
case of the affine plane and we show the role of theorems by Desargues and
Pappus in the coordinatization of the plane. This exposition highlights the
connection between the geometrical properties assumed as axioms of the
algebraic properties. In the second paper, which is an ideal sequel to the
previous one, we illustrate the procedure that leads to identifying the
equations of the affinities on the basis of merely geometrical considerations.
149
FIORI C., ZUCCHERI L.
I numeri reali il continuo aritmetico: quale conoscenza alla fine degli studi
universitari per i futuri insegnanti? ('Real numbers and the arithmetical
continuum: how much do future teachers know by the end of their university
studies?'), L'Educazione Matematica, (Oct 1996) v. 17(3) p. 158-174.
school level: e, m; mathematical subject: ar; educational area: d.
Interviews with students finishing a degree course in mathematics (mostly
future teachers) and seminars with high school teachers have on a number of
occasions shown how difficult it can be to explain the system of real numbers,
especially as regards aspects concerning its properties of completeness and
the notion of a non numerable infinity. Starting from this consideration, the
authors have carried out an investigation for determining what level of
knowledge of real numbers, the students have by the end of high school and at
the end of a degree course in mathematics. In order to carry out the research,
the authors submitted a questionnaire to the students who were also
individually interviewed; in this paper they present the more significant
results that emerged from the qualitative and quantitative analysis of the
answers given. The results shown that the algebraic aspects are prevalent and
therefore it seems to be necessary, in the teaching, to treat in a deep way the
other aspects, i.e. the cardinality, the ordinal structure and the associated
topological questions. The paper is devoted to teachers of university and of
secondary school.
FIORI C., ZUCCHERI L.
Errori nell’applicazione dell’algoritmo della sottrazione: un’analisi relativa
alla scuola dell’ obbligo (‘Errors in performing the subtraction algorithm: an
investigation in primary and middle school’), L'Insegnamento della
Matematica e delle Scienze Integrate, 1997, 20A(1) 7-38
school level: e;
mathematical subject: ar; educational area: pcb.
In the usual teaching of maths, pupils’ mistakes are often evaluated from the
negative point of view, while on the contrary, if suitably analyzed, they may
give useful suggestions for improving the teaching/learning process. In this
paper we present the results of an investigation on a population of 732 pupils
(9-12 years old), addressed to determine the typology of errors in performing
the (written) subtraction algorithm. The results, as well as the corresponding
analysis, are then compared with those obtained at the University of Belo
Horizonte (Brazil). We finally conclude with some didactical considerations
about the use of two different algorithms for the (written) subtraction.
150
FURINGHETTI F.
History of mathematics, mathematics education, school practice: case studies
linking different domains, For the learning of mathematics, 1997, vol17, n.1,
55-61.
school level: e, m, b, t; mathematical subject: hs;
educational area: tr.
In this paper we consider how the domains of mathematics education and
history of mathematics may interact in the process of mathematics teaching.
We try to tackle the problem focusing on the teachers’ role, which is central
in this process. The basic point is that the use of history needs of the
epistemological and cognitive analysis. Through the discussion of specially
chosen examples we try to outline a cognitive model which explains the role
of history in learning mathematics. We also give a classification of the types
of use which can be made of history, according to the teachers’ aims in their
classroom work.
FURINGHETTI F.
Mathematics teacher education in Italy: a glorious past, an uncertain present,
a promising future, Journal of mathematics teacher education, 1998, vol.1,
341-348.
school level: u; mathematical subject: m; educational area: tt.
In this article we analyze aspects of the professional life of Italian
mathematics teachers with particular reference to their education. The main
problem emerging from our analysis is to find a balance between the subject
matter knowledge and the pedagogical matter knowledge. In the past there has
been some attention for the pedagogical/ psychological education of teachers.
Afterwards the community of mathematicians pushed towards a strong
mathematical education, without place for educational issues. We analyze
recent activities in the field of teachers retraining which seems oriented to
regain the lost balance of the types of knowledge (educational and
mathematical) for teaching.
FURINGHETTI F., PAOLA D.
Shadows on proof, in Proc. PME 21 Lahti, Finland, 1997, v.2, 273-280.
school level: b, t; mathematical subject: ar; educational area: p.
In this paper we refer to an experiment in which students of the age range 1417 have to proof a statement on natural numbers, writing all their thoughts
151
while they are working on this task. We perform a kind of ‘genetic
decomposition’ of the statement and single out some parameters, on which we
base the analysis of the students’ protocols. The main schemes found in
students’ proofs are the authoritarian, the empirical, the ritual and the
symbolic. We study the relations of these proof schemes with the context
chosen by the students. Some students’ behaviors allow to single out elements
suggesting the influence of the algebraic or arithmetic contexts on proving
this type of statement: we call it algebraic or arithmetical shadow effect.
FURINGHETTI F., PAOLA D.
Context influence on mathematical reasoning, in Proc. PME 22, Stellenbosch,
South Africa, 1998, vol.2, 313-320.
school level: b; mathematical subject: l; educational area: p.
The problem studied in this paper is how and under which conditions students
accept or refuse the rules of formal deduction. In particular, the focus is on
the role of the context in the activity of proving, where by ‘context’ we mean
the ‘semantic context’ of the statement to be proved and not the global
context in which the classroom is set. Our study is based on the analysis of
the answers of 40 students aged 16 years to a questionnaire on the
introduction and elimination of ‘and’, and on the introduction of ‘or’. The
results of our analysis reveal, in our opinion, a remarkable interference of the
context, which includes both the semantic meaning of the propositions
involved in a deduction step and certain implicit assumptions induced by the
common usage of certain words in the natural language; this is particularly
evident in the case of the introduction of or.
FURINGHETTI, F., PAOLA, D.
Exploring students’ images and definitions of area, in Proc. PME 23 Haifa,
Israel, 1999, vol. 2, 345-352.
school level: t;
mathematical subject: g; educational area: p.
This study examines several aspects of the images and definitions that eight
students of high school (16 years old) have regarding area. The analysis is
performed through 10 open questions to which students answered through
written statements, drawings, concept maps. The protocols shed light on the
ways used by students to communicate their ideas and on the role they ascribe
to definitions in their mathematical experience.
The attention for problem linked to the use of logic in activities of proving is
present in other studies we have carried out, see for one
CICERI, C., FURINGHETTI, F. & PAOLA, D.: 1996, Analisi logica di
dimostrazioni per entrare nella logica della dimostrazione [Logical analysis of
152
proofs to enter into the logic of proof], L’Insegnamento della Matematica e
delle Scienze Integrate, v.19B, 209-234.
FURINGHETTI F., PAOLA, D.
Parameters, unknowns and variables: a little difference?, in Proc. PME 18,
Lisboa, Portugal, 1994, vol. 2, 368-375.
school level: t;
mathematical subject: al; educational area: tr.
In this paper we report on a research concerning algebra learning in secondary
school; the focus is on parameters and their relation to unknowns and
variables. In developing our study we at first analyzed the notion (in its
manipulative and conceptual aspects) using a methodology we had already
tested in other studies on algebra learning which consists in singling out what
lies behind to a given notion and in constructing a tree of notions related to
the initial one. We then prepared a questionnaire to establish how students
perceive the differences between parameters, unknowns and variables and
deal with algebraic situations where these notions intervene. The
questionnaire was handed out to 199 students aged 16-17 of 3 schools. The
results offer us useful insights for the analysis of fundamental aspects of
algebraic thinking. This paper has to be integrated with the following paper,
in which the problem of parameters is taken from the point of view of
teaching prescriptions and textbooks:
CHIARUGI, I., FRACASSINA, G., FURINGHETTI, F. & PAOLA, D.: 1995,
‘Parametri, variabili e altro: un ripensamento su come questi concetti sono
presentati in classe’, L’Insegnamento della Matematica e delle Scienze
Integrate, v.18B, 34-50.
FURINGHETTI F., SOMAGLIA A. M.
History of mathematics in school across disciplines, Mathematics in school
(History of mathematics - Extra special issue), 1998, vol.27, n.4, 48-51.
school level: t;
mathematical subject: hs;
educational area: tr.
One of the problems in classroom life is the phenomenon that we can term the
'fragmentation of knowledge', due to the fact that school subjects are taught
separately from each other so that the result is islands in the learning process.
In particular, mathematics, which is difficult for many pupils, suffers from
this situation and, more than other subjects, is considered to be separated from
the cultural context. As a result, the image of mathematics held by pupils is
very poor: pupils think that mathematics is a very boring subject, without any
imagination, detached from real life. In this paper we analyze the role that
history may have in modifying this attitude of students. In particular, we
153
examine the character of history to go across disciplines in examples
developed in classroom.
GALLO E.
Algebraic manipulation as problem solving, Proc. First Italian-Spanish
Research Symposium in Mathematics Education, Modena, 131-138
school level: b; mathematical subject: al; educational area: d, mr.
In a preceding paper Control and solution of “algebraic problems” we
studied with 14/15-years-old students the control dynamics during the
solution of algebraic exercises in order to create an ad hoc model for the
solution. The aim of this research is to use the dynamics of ascendent and
descendent controls to highlight the models which are brought into play by
subjects during the formal manipulation which accompanies the solution of a
normal algebra exercise, assumed as a problem, and the role of these models
in relation to the two levels of formal ostension and conceptualisation.
By focusing special attention on the importance of algebraic manipulation,
our position is aligned with those who consider there to be “room for
discussion and research on how proficient students in today’s technological
world need to be in manipulative skills” given that today it is important for
algebra students “to attain a high level of competence in symbol
manipulation” (Thorpe).
Our decision to focus attention on the construction of specific models and on
the dual descendent and ascendent process, brings us into line with the
research on genetic psychology carried out by the Geneva group (Saada
Robert).
GALLO E.
Représentations en géométrie et en algèbre: une confrontation
('Representations in Geometry and Algebra: a comparison'), Proc. CIEAEM
46, 1994, 256-263
school level: b; mathematical subject: al, g;
educational area: tr
During the solution of geometrical problems or algebraical exercises the
subject constructs representations of the situation; we compare such
representations by presenting the different frames of reference used in our
researches to interpret students’ productions (14/15-years-old).
The representation of the situation makes the student uses previously acquired
knowledge, namely models, sometimes transforming it to build specific
knowledge, other models, which are relevant to his problem or exercise: we
take account of the degree of model formation (structuring) and of the degree
154
of its appropriateness (applicability) to the requirements of the situation, as
done by Ackermann-Valladao in a research on models formation and
actualisation in problem-solving. When we centre our attention on the stages
of the passage between one model and another, problem resolution appears to
be made up of a succession of interpretative cycles: the meaning of the
situation for the student plays a very important role to create a good
representation for the resolution.
GALLO E., TESTA C.
Comment peut-on analyser le discours de l’enseignant en classe?' ('How can
teacher's discourse within the classroom be analysed?'), Proceedings of the
CIEAEM 50, (Neuchâtel Switzerland,1998) 1999, 407-410
school level: b, t; mathematical subject: m; educational area: m.
We suggest a method which is based on three complementary analyses of the
teacher’s speech: linear analysis, category analysis, longitudinal analysis. The
linear analysis consists of the linear description of the activities which are
carried out in the class, divided into units so that each unit corresponds to
only one activity for the students. The category analysis consists of the break
up of the teacher’s speech into strictly mathematic speech and accompanying
speech, broken up into three categories: management, labelling, reflection.
The longitudinal analysis consists of pointing out the possible invariants of
the teacher’s speech. Linear, category, longitudinal analysis are words used
by A. Robert and J. Robinet in their research.
Transversal to the three analysis is the theme of intentionality in teaching:
intentionality is linked with intentio (intention of school system) and with
didactics intentions (the intention accomplished in class). By focusing special
attention to intentionality our position is aligned with Portugais’ studies.
GALLO E., BERTONI, V., SACCO M.P., MARCHISIO S., TANZI
CATTABIANCHI M.
L’omotetia a livello elementare e medio: aspetto figurale e concettuale.
Un’analisi in termini di modelli' ('Homothety at the level of primary and
secondary school: figural and conceptual aspects. An investigation in terms of
models'), Grugnetti L & Gregori S. (eds.) Dallo Spazio del bambino agli spazi
della geometria, Atti del 2° Internuclei Scuola dell'Obbligo, 1997, 11-20
school level: e, m; mathematical subject: g; educational area: d, mr.
In Euclidean geometry similitude is introduced as comparison between
figures, not as transformation. When one wants to teach similitude from the
second point of view, homothety is a useful and necessary way of starting and
155
of moving from the intrafigural phase to the interfigural. By focusing special
attention on the passage from the first phase to the second, our position agrees
with the conclusions of J. Piaget and R. Garcia in studies made in a
epistemological framework. In this paper we present two learning plans, one
in primary school (here control based on the alignment of corresponding
points with the homothety centre is perceptive and instrumental) another in
lower secondary school (here control brings to the conceptual construction of
homothety). This work is an example of “didactical engineering” since we
study didactical performances in class using research methodologies (M.
Artigue).
GALLOPIN P., ZUCCHERI L.
Fare geometria col solo compasso utilizzando Cabri (Doing geometry with
compasses only using Cabri'), La Matematica e la sua Didattica, Gen-Mar
1999, n. 1, p. 98-123.
school level: b; mathematical subject: g; educational area: cm.
The Mascheroni Theorem assures that any geometrical construction by ruler
and compasses can be also carried out using compasses only. The authors
propose a didactical itinerary for 15-16 years old pupils, which leads step by
step to the proof of this theorem, and they suggest further how these
constructions can be realized by means of the software Cabri. In the
following, they underline the more relevant didactic aspects related with the
use of the tool. The aim of the proposed activity, from an educational point of
view, is to stimulate the pupils to a deeper analysis of the fundamental
geometrical concepts that they have already learned, but that they generally
use in mechanical way.
GARUTI R.
A classroom discussion and a historical dialogue: a case study, in Proc. PME
21, Lahti, Finland, , vol. 2, pp. 297-304
school level: m; mathematical subject: hs;
educational area: p, pcb.
This report deals with a comparison between a mathematical discussion in the
classroom and an historical dialogue. Both regard the mathematical modeling
of the phenomenon of the fall of bodies and in particular the possible
dependence of the fall speed on the traversed space. The protagonists of the
classroom discussion are 8th grade students, while the protagonists of the
historical dialogue are Simplicio, Sagredo and Salviati (Galilei, 1638).
Analysis and comparison of the two 'discussions' raises issues concerning:
interpretation of the analogies between them; and the conditions that allowed
156
the classroom discussion rapidly to cover some important steps in the
development of scientific thinking represented in the historical dialogue (this
was read after the discussion!).
GARUTI R.; BOERO, P.
Mathematical Modelling of the Elongation of a Spring: Given a Double
Length Spring .....', Proc. PME 18 , 1994, vol. 2, pp. 384-391.
school level: m; mathematical subject: m; educational area: mr.
This report concerns mathematical modelling in mathematics education. It
analyzes how, in facing a specific problem of mathematical modelling, the
students of two classes of grade VIII have used resources such as their
conception of the phenomenon, the mathematical tools available, their
previous modelling experience as well as some general "principles". This
report also compares two different ways of managing the operations of
verification of the modelling hypotheses produced.
GARUTI R.; BOERO, P. LEMUT E.
Cognitive Unity of Theorems and Difficulty of Proof, in Olivier, A. & Karen
Newstead (eds.), Proc. PME 22, Stellenbosch, South Africa, 1998, vol.2, 345352
school level: m; mathematical subject: m; educational area: p.
The cognitive unity of theorems - a theoretical construct originally elaborated
to interpret student behaviour in an open problem solving holistic approach
to theorems - has been transformed into a tool that may be useful for
interpreting and predicting students' difficulties when they are engaged in
proving statements of theorems. The aim of this paper is to explain (through
"emblematic" examples) the potentialities of this tool and indicate possible
further developments concerning both research and educational implications
for the approach to proof in schools.
GARUTI R., BOERO P., CHIAPPINI, G.
Bringing the Voice of Plato in the Classroom to Detect and Overcome
Conceptual Mistakes, Proc. PME 23, Haifa, Israel, 1999, vol. 3, pp. 9-16
school level: e, m; mathematical subject: m; educational area: m, p.
The capacity of detecting conceptual mistakes and overcoming them by
general explanation is important in the approach to theoretical knowledge,
and its development in students calls for the teacher's intervention. Our
157
working hypothesis is that the "voices and echoes game" can function as an
appropriate methodology to this end. In order to explore this perspective in
depth, a teaching experiment was performed in six classes (grades V and
VII). This report provides a partial account of this complex experiment,
presents some results and highlights some open research questions.
GRUGNETTI L.
Il concetto di funzione: difficoltà e misconcetti (Difficulties and
misconceptions of the concept of function) in L'Educazione Matematica,
Anno XV - Serie IV - Vol. 1, n. 3, 1994, 173-183.
school level: b, t; mathematical subject: c; educational area: pcb.
A version of this paper was presented at the Fifth International Conference on
Systematic Cooperation between Theory and Practice in Mathematics
Education, Grado (Italy, May, 23-27, 1994, L. Bazzini, ed. of the
proceedings). In this bilingual article, a study on a sample of 102 students of
the first year of the faculty of engineering of the University of Parma is
presented in order to analyse and to put into evidence an inadequate
understanding (or a long term assimilation) of the concept of function by
students. Theoretical aspects "responsible" of difficulties of this concept are
considered taking into account also the historical development of the concept
of function. The different names that this concept assumes - operation,
correspondence, relation, transformation - reflect the historical circumstances
in which it appeared on the fields of mathematics, of physics, of logic.
Advantages and disadvantages in teaching the concept of function by using its
different interpretations (or descriptions) are considered.
GRUGNETTI L.
Relations between history and didactics of mathematics, in Proc. PME 18,
Lisbon, Portugal, 1994, vol. I, 121-124.
school level: b, t; mathematical subject: hs;
educational area: tr, e.
In this paper, potentialities, but also risks and limits of history of mathematics
in mathematics education are taken into account and analysed. With regard to
mathematics, an historical approach could be advantageous for the students
because it allows them to think of mathematics as a continuous effort of
reflection and of improvement by man, rather then a "definitive building"
composed of irrefutable and unchangeable truths. But, once we decide to
follow this approach, it is important to avoid the risks of falling into
anachonism, and of increasing a notionistic view, of giving to students a
fragmentary idea of the history of mathematics. A historical approach to
158
mathematics involves the passage from a disciplinary to an interdisciplinary
treatment in the broadest sense of the word. Examples in this sense are
given.as well as a survey of Italian experiences in this field.
GRUGNETTI L.
Pensiero proporzionale e costruttivismo: superamento di ostacoli?
(Proportional thinking and constructivism: obstacles overcome?), in Grugnetti
L & Gregori S. (eds.) Dallo Spazio del bambino agli spazi della geometria,
Atti del 2° Internuclei Scuola dell'Obbligo, 1997, 77-82.
school level: m, b; mathematical subject: ar; educational area: d
This paper presents research on difficulties in proportional reasoning which is
also important in geometry. Despite much interesting research in this subject,
several crucial questions concerning the mastery of proportional reasoning
remain. This research deals with the aim of studying the relationship between
socio-costructivism and overcoming some obstacles - in particular the power
of additive structure - in proportional reasoning and in recognising a
proportional problem.
GRUGNETTI L., JAQUET F.
Senior Secondary School Practices, in A.Bishop et al. (eds) International
Handbook of Mathematics Education, Kluwer Academic Publ., 1996, 615645.
school level: b, t; mathematical subject: m; educational area: d, e.
This paper deals with senior secondary school practices, which for most
systems means the teaching of students aged between 15 and 18 years. This
sector of mathematics education has come under increased scrutiny as more
students are staying on after their junior secondary school time and as access
to higher education has developed dramatically over the past decade. Rather
than attempting to survey world-wide practices, an impossible task in itself,
the authors have chosen to focus on four topics with which to analyse the
trends, developments, and issues. The four topics are:
1) problem solving
2) the evolution of mathematics teaching objectives and practices
3) new tools for calculating and representing functions
4) the contribution of the epistemology and the history of mathematics.
GRUGNETTI L., SPERANZA, F.
159
Teacher Training in Italy: the State of Art, in N: A: Malara and L.Rico (eds)
Proc. first Italian-Spanish Research Symposium in Mathematics Education,
Modena (Italy), 1994, 205-210.
school level: u; mathematical subject: m; educational area: tt.
The authors present in this paper a synthetic outline of the problems and the
reality of the situation in Italy of teacher training from primary to hight
school. The "practicability" of a new law (in Italy) that prescribes a university
training for primary teachers and a post-graduate courses for secondary
teachers is analysed starting from a historical survey of Italian laws in this
field. In effect the "Italian case" has some noteworthy characteristics,
depending on the history of Italian culture.
GRUGNETTI L., SPERANZA, F.
General reflections on the problem history and didactic of mathematics: Some
answers to the Discussion Document for the ICMI Study on the role of the
history of mathematics in Philosophy of Mathematics Education Journal 11
(1999)
school level: e, m, b, t, u;mathematical subject: hs;
educational area: e,
d.
In this paper the authors give some answers to the Discussion Document for
the ICMI Study on the role of the history of mathematics taking into account
the Italian cultural situation and point of view. For each of the ten questions in
the Discussion Document, comments coming from the authors' researchs on
history and epistemology of mathematics in mathematics education are given.
The importance of these aspects is stressed, but also dangers and difficulties
are put into evidence.
GRUGNETTI L., JAQUET F., VIGHI P.
Rally matematico alla scuola elementare (Mathematical 'rally' in primary
school), in L'Educazione Matematica, Anno XVI - Serie IV - Vol. 2, n. 3,
1995, 113-123. (first part) and Anno XVII - Serie V - Vol. 1, n. 1, 1996, 1-12.
(second part)
school level: e, m; mathematical subject: m; educational area: d, mr, pcb.
This bilingual paper (in two parts) deals with a problem-solving activity for
primary school within the context of a mathematics rally, that is a classes'
competition . In the first part, the educational aims of such an activity and the
quality of the terms of the problems are analysed and some examples of
problems are given.
160
In the second part, 'a priori' and 'a posteriori' analysis are proposed, compared
and discussed. In particular, the stategies adopted by the pupils, the evaluation
of their work and the influence of the word statement of the problems on their
strategies are analysed.
GRUGNETTI L., RIZZA A., BEDULLI M., FOGLIA S., GREGORI S.
Le concept de limite: quel rapport avec la langue naturelle? ('The concept of
limit: which relationships with natural language?), in F. Jaquet (ed.)
Relationships between classroom practce and research in mathematics
education, Proc. CIEAEM 50 (Neuchatel, Switzerland, 1998), 1999, 313-318.
school level: b, t; mathematical subject: ac;
educational area: pcb, e, d.
In the paper, the first step of a study concerning the learning of the concept of
limit, is presented. One of the obstacles in the understanding of this concept
and its implications in studying Calculus are analysed. Among the numerous
aspects, more or less well known, that are related to the learning of the
concept, our work pointed out the importance of the linguistic aspect. The
assumption that natural language affects or even hinders the understanding
and the acceptance of the mathematical concept of limit was confirmed. In
particular the “strong” idea of limit as barrier and the deriving negative
connotation can represent a huge pre-existing obstacle to any didactic action.
Such an idea, together with the well known epistemological difficulties,
makes the teacher’s efforts hardly effective. The examination of carried out
enquiries has pointed out that the mastery of calculation techniques does not
always coincide with an actual understanding of the concept in students.
GUALA E., BOERO P.
Time Complexity and Learning, in Tempos in Science and Nature: Structures,
Relations and Complexity, vol. 879, Annals of the New York Academy of
Sciences, pp. 164-167
school level: e, m, b, t, u;mathematical subject: m; educational area: e.
The debate about the physical existence of time suggests the possibility that
time could also be considered as an intellectual construction in order to "treat"
(that is, to describe/ order/ analyse) the flow of external events; in addition, it
raises the problem of intellectual constructions suitable for "treating" the flux
of internal events. On this point, we can speak about "mind times",
metaphors which may help in "treating" mental processes, especially those
intervening in complex problem solving. In this paper we consider in a
phenomenological manner the variety of "times" that the mind must manage
in mathematical problem solving. We also consider the intertwining amongst
161
them, mentioning some examples (in which success or failure seems to
depend on the capacity to manage such time complexity). Finally, we
consider the hypothesis that the analysis of "mind times" may be useful (in an
"embodied cognition" perspective) for singling out some mental processes on
which basic mathematical ideas and skills are founded.
LANCIANO N.
Aspects of teaching learning geometry by means of astronomy, in Malara N.,
Rico L., Proc. First Italian Spanish Symposium in Mathematics Education,
AGUM, Modena, 43-49
school level: e;
mathematical subject: g; educational area: pcb.
In this paper I present a part of a wider study carried out on children aged
between 6 and 11. In the first section I explain what I mean by research on
learning and teaching in this context. I refer mostly to the theoretical outlook
of André, Giordan and his co-workers at the LDES in Geneva and within this
framework I consider a few terms, in some cases adapted to the subject under
discussion, and use them in describing my research. Then I explain what I
mean in this context by Geometry and Astronomy. Finally, I give some
examples of processes of learning and teaching in these contexts.
LEMUT, E., GRECO, S.
Re-starting algebra in high school: the role of systemic thinking and of
representation systems command', Proc. PME 22, Stellenbosch, South Africa,
1998, vol. 3, pp.191-198
school level: b; mathematical subject: g, al;
educational area: tr, mr.
In this paper we focus on restarting algebra in the first years of high school; in
particular, we analyse connections between ability of thinking in systemic
terms and algebraic modelling and discuss about the significant influence of
students attitude and capability to make use of representation registers rich of
operative potentialities on ability of algebraic modelling a situation; we
finally suggest some didactic implications of our analysis on classroom
activities.
LEMUT E., MARIOTTI M.A.
Pictures and Picturing in Elementary Problem Solving, Proc. European
Research Conf. on the Psychology of Math. Education -Osnabruck, 1995,.4245
school level: e;
mathematical subject: g; educational area: mr, v.
162
In this paper we analysed the role of pictures and picturing in solving some
problems concerning a spatial situation. In our analysis we met pictures in
different subjective roles: illustration, validation, modelling. We analysed
essentially the role of modelling, considering a picture as a model if it shows
structured information, and a good model for the pupil if it can suggest
specific which lead to the achievement of a solution.
LETIZIA A., MARCHINI C., IACOMELLA A
Logic for assessing or assessment in Logic, Proc. of the CIEAEM 50,
(Neuchatel Switzerland 1998), 1999, 319 - 323.
school level: b; mathematical subject: l; educational area: pcb, tt, va.
This paper starts from the book: Iacomella, A. and Letizia, A. and Marchini,
C.: 1997, Il progetto europeo sulla dispersione scolastica: un'occasione di
ricerca didattica - Dalla lingua d'uso comune e con il buonsenso verso l'idea di
connettivo logico e di quantificatore logico, (European project on school drop
out: an opportunity for research in mathematical education - From colloquial
language and common sense towards ideas of logical connective and
quantifier), Editrice Salentina, Galatina., originated from the problem of
school drop out. This phenomenon has many social and economic
components; we address to secondary school teachers, emphasizing some
aspects of the everyday-life language. In our experience students aged 14
show incomprehension of colloquial terms, these misunderstandings block
many school activities and interfere with the learning. Our purpose is to
redirect the teacher's attention to the formation of disciplinary languages
starting from a deeper analysis of phrases. We stress the importance of
semantic, syntactic and morphologic aspects of the language and the use of
specific vocabulary and context, giving meaning to these phrases. In the paper
we stress that Logic is a tool used to assess: the teacher must recognize in an
assessment how many disciplinary contents there are and also the quality of
the organization of the contents. Logic is used in these qualitative aspects of a
quantitative assessment (Logic for assessing). In the paper we present a sort
of grid, by focusing some aspects of the interplay capacities/abilities, on
typical logical contents starting from common use language. We present a
classroom work organized in many steps, which lead to the achievement of a
classical first order predicate language. Each step is presented with proper
proofs suggested for assessment in Logic.
LLADO' C., BOERO P.
Les intéractions sociales dans la classe et le role médiateur de l'enseignant
dans la modélisation mathématique des phénomènes naturels: le cas de la
163
génétique' ('Social interactions in the classroom and the teacher's mediation
role in mathematical modeling of natural phenomena: the case of genetics'),
Proc. of CIEAEM-49, (Setubal, Portugal, 1997), 1998, 171-179
school level: m; mathematical subject: p; educational area: ap.
The aim of this paper is to explore the possibility of provoking the evolution
of students' conceptions about the phenomenon of heredity through different
types of interactions between teacher and students and between students. This
study concerns the approach to Mendel's probabilistic model. It is based on a
teaching experiment performed in grade VII. The "game of hypotheses" and
the "voices and echos game" are key theoretical issues.
MALARA N.A.
E’ possibile limitare le difficoltà in matematica e farla apprezzare agli allievi?
(Is it possible to limit the difficulties of mathematics and make pupils like
it'?), Insegnamento della Matematica e delle Scienze Integrate, 1995, vol.
18A/B n. 5, 551-570,
school level: m, b; mathematical subject: m, al; educational area: i, tcb.
This paper consists of two parts. The first contains a reflection on today's
image of mathematics, with a general overview of what has created it, as well
as an analysis of the new challenges that society gives to school, of the role
this discipline has today, of the teachers' role and training. The second part
contains a synthetic report of the activities carried out with and for middle
school teachers in order to promote a more appropriate image of mathematics,
more respectful of its peculiarities and of the effects of such work in the
pupils. This report shows how it is possible, with a purposed methodology, to
make pupils overcome the traditional beliefs about mathematics that are at the
basis of the social prejudices and of the damage against its image.
MALARA N.A
Mutamenti e permanenze nell'insegnamento delle equazioni algebriche da
un'analisi di libri di testo di Algebra editi a partire dal 1880 (What has
changed and what remains unchanged in the teaching of algebraic equations,
according to a compared analysis of algebra textbooks published 1880 to
now), in BAZZINI L. (ed.) La Didattica dell'Algebra nella Scuola secondaria
Superiore, 1997, ISDAF, Pavia, 145-154
school level: b; mathematical subject: al educational area: cr.
This paper concerns a study of ancient and recent texts of algebra for upper
secondary school, aimed at highlighting what has changed and what has
164
remained unchanged in the didactics of algebraic equations as it emerges from
the compared analysis of Italian relevant school textbooks published since
1880. We summarize the results of the analysis of each book examined as to
the following points: introduction to algebraic equations and principles of
transformation; equations of first degreewitht one unknown; equations of
second degree with one unknown; equations with one unknown of degree
higher than the second; rational fractional equations; irrational equations;
equations with literal coefficients or with parameters; equations with two or
more unknowns; problems and equations, systems of linear equations;
systems of non-linear equations. We noticed a significant change in
perspective between ancient and recent texts: from a deductive, abstract and
general approach to an inductive one, based on the graphical-geometrical
model; we also detected a change in the kind of practice activities for the
students, now less complex from a technical-operational point of view.
Moreover, these texts show a progressive deterioration from the algebraic
cultural point of view.
MALARA N.A. (ed)
An International View on Didactics of Mathematics as a Scientific Discipline,
1997, AGUM, Modena
school level: -;
mathematical subject: m; educational area: d, e, tr.
This book contains the proceedings of the Working Group "Didactics of
mathematics as a Scientific Discipline" in ICME 8 (Seville, 1996). The goal
of the working group was to create a forum to highlight the status of didactics
of mathematics as a scientific discipline, focusing on its objects and core, on
its connections with other fields (epistemology, anthropology, sociology,
psychology, etc.) and on its features in the various cultures, in order to
achieve an internationally shared vision of it.
The book consists of three parts and an appendix. The first part concerns the
presentations of the subgroup devoted to the objects and core of didactics of
mathematics as a scientific discipline, to the influences from connected
disciplines and today's trends, and contains, among others, interesting studies
by: Gascon (Spain), Lerman (Great Britain), Marafioti (Brasil), Mousley
(Australia), Safuanov (Russia) e di Speranza (Italy). The second part concerns
the presentations held in the subgroup aimed at framing the status of
mathematics as a scientific discipline in the various countries, among which
we pinpoint the interesting surveys by Gelfman & al. (Russia), Iwasaki
(Japan), Sowder (USA). It contains also a paper of ours, written in
collaboration with M. Menghini, in which we trace the guidelines of Italian
research as they arise from the analysis of the book Italian Research in
165
Mathematics Education: 1988-1995, which we edited for this occasion
together with M. Reggiani. The third part contains interesting contributions
offered to the round table of the working group by Lerman (GB), Pellerey
(Italy), Silver (USA), Wittmann (Germany) and a further contribution by F.
Speranza, as a reflection on the issues emerged in the discussion, hand in
hand with Wittmann's ideas.
The image of mathematics resulting from the book is not unitary. The various
contributions show a wide range of conceptions, from the mainly theoretical
to the mainly practical, and they all reflect the social, cultural and historical
conditions of the country in which they have developed. The discipline
appears to be quite consolidated in America (both in the USA and in South
America, even if with a more practical nature in the North, probably owing to
the influence of the Anglo-Saxon culture, and a more theoretical nature in the
South, owing to the possible influences of the Spanish-Portuguese culture),
whereas in Russia and Japan it seems to be less autonomous from the social
and political structures, although it tends to be rather oriented towards
speculation. As to the Italian image of didactics of mathematics as a scientific
discipline, it seems rather fluid: most of its studies move mainly from the
praxis and are praxis-oriented, still there are some interesting evolutions
towards more theoretical aspects as well as studies of modelization of the
teaching-learning processes.
MALARA N.A.
On the difficulties of visualization and representation of 3D objects in middle
school teachers, Proc. PME 22 1998, vol. 3, 239-246.
school level: m; mathematical subject: g; educational area: tt, v.
This paper reports the results of an experience carried out within a teachers'
seminar devoted to innovation in teaching 3D geometry. It concerns the
theachers’ behaviour and difficulties on studying some questions about the
representation of solids on isometric paper under certain conditions, which
requires the ability to visualize the effects of some shifting of solids or to
evoke the vision of objects from particular points of view, as well as to
represent them correctly. We describe the goals and the difficulties foreseen
for each exercise, moreover we examine the teachers' productions, with
particular attention to those which show mistakes and witness uncertain,
unforeseen or hardly imaginable visions. We conclude with some
considerations about the opportunity, the strategies and the timing for
introducing these and other activities of solid geometry belonging to the
project, and also considerations about how to improve the teachers'
competence in this field.
166
MALARA N.A.
An aspect of a long-term research on algebra: the solution of verbal problems,
proc. PME 23, 1999, Haifa, Israele, vol. 3, 257-264
school level: m, b; mathematical subject: al;
educational area: cr, tt.
This paper collects the results of an experiment carried out in a second grade
class concerning the solution of algebraic word problems. It deals with the
pupils' difficulty of translating text information into algebraic language and
managing its elaboration, by facing the syntactical questions that gradually
arise along the activity. This research has highlighted how the pupils, when
properly guided, can represent verbal relations in different ways, compose
relations by substitution, achieve the solution of problems with more than one
unknown without any specific study of syntactical kind and be aware of the
need and importance of studying expressions and algebraic equations
autonomously.
This is also the approach of the study MALARA N.A., NAVARRA G., 2000,
Explorative ways to encourage algebraic thinking through problems, in
Gagatsis, Makrides G. (eds) proc. II Mediterranean Conference on
Mathematics Education, vol 1, 55-64, concerning a teaching experiment of
solution of algebraic problems between elementary school and middle school,
aimed at approaching the concept of linear equation. This research sees
equations as the final point of a process activity focused on progressive
schematization of representations of situations concerning the use of a twopan balance, , which allows that pupils to concentrate their attention on the
equivalence principles as "in-progress theorems".
MALARA N.A., GHERPELLI L.
Problem Posing and Hypothetical Reasoning in Geometry', Proc. PME 18,
Lisbon, Portugal, 1994, vol. 3°, 216-224.
school level: m; mathematical subject: pg;
educational area: cr.
We expose the guidelines and the main results of a research carried out with
12/13-year-old pupils, aimed at leading them to posing problems within
elementary plane geometrical figures by constructing problem texts by
themselves. The aim of this research was to investigate the actual possibilities
for pupils of this age to construct problems within the field considered and to
gain information about the effects of co-operation on the topic. This research,
divided into three stages, respectively saw the pupils as producers of problem
texts, critical revisers of the problems created and observers of analogies with
other problems created and/or problems in books. Generally speaking, we
found out that problem posing fosters the development of problem-solving
167
abilities under hypothesis (by identifying and developing possible problems
related to a given geometrical figure) and promotes metacognition (through
the control of the strategies underlying the various situations constructed, the
awareness of the fundamental relationships among the elements of the
geometrical figures considered and the control of the range of the classical
models of problems related to them). With regard to methodology, we
detected the effectiveness of group activity both for producing/solving
problems and for overcoming weaker pupils' difficulties.
The wider study MALARA N.A 1994, The problem as a Tool for the
Promotion of Hypothetical Reasoning and Metaknowledge, in Bazzini L.,
Steiner H G. (eds) Proc. Second Italian-German Symposium on Teaching of
Mathematics, 303-324 is at the basis of such activity. This study is a synthesis
and a reflection on the results of the researches the author had carried out on
the didactic of "the problem", with particular reference to the argumentative
and metacognitive skills developed by the pupils as to: a) logical problems of
various kinds (combinatory, relational, true/fals type, etc.); b) rough-state
problems taken from situations close to pupils' experience; c) construction of
problem texts by the pupils, either in an arithmetical or in geometrical field.
Another study aimed at promoting a teaching focused on the control of pupils'
solving strategies is reported in MALARA N.A NAVARRA G., 1998, Role of
the teacher in promoting interaction among pupils and metacognition through
problem solving abilities , proc. CIEAEM 49, 203-211, in which we also
linger on the role of the teacher and on the crucial importance of his/her
action on these aspects.
MALARA N.A., IADEROSA R.
Difficulties met by pupils in learning direct plane isometries,, Proc. PME 21,
Lathi, Finland, 1997, vol. 3, 208-215
school level: m; mathematical subject: pg;
educational area: cr, tt.
This paper can be considered as the concluding study of a long research
programme for didactical innovation on plane isometries, realized through the
use of the computer. It focuses on some difficulties met by pupils in the
mental representation and in the conceptualization of plane isometries as
mathematical objects. The hypothesis of the research was that the dynamic
visualization of the action of a geometrical transformation on various figures,
not necessarily convex or limited, and on sets of loose points, can lead the
pupils to: a) construct the appropriate mental images for overcoming wellknown difficulties they have on realizing the correspondent of figures
according to a certain isometry; b) achieve the meaning of invariant and unite
element in a transformation and arrive at the concept of this as a
correspondence between points of the plane. The research has shown that,
168
even if computer visualizations allowed the pupils to achieve a good inner
vision of classes of figures united by translation or rotation, several of them
had conflicts in representing the correspondent of a translation of a right line
according to a vector parallel to it, or in realizing the correspondence of a
certain pair of figures, such as a circle and an a right line tangent to it,
according to particular translations or rotations. Moreover, as to the extension
of the transformation to the whole plain, several pupils showed the persitence
of a local vision .
Also the paper MALARA N.A., IADEROSA R.1995, How much does "common
sense" influence the teacher's ability in recognizing pupils' difficulties? proc.
CIEAEM 47, 361-369, refers to this research. It reports the teachers' visions
on the goals of the worksheets created for this topic and on the pupils'
difficulties foreseen. As a background to these researches there is the paper
PINCELLA M.G., MALARA N.A.: 1995, The informal study of transformations
and invariants as an approach to geometry in middle school, La Matematica e
la sua Didattica, n. 4 , 446-462, in which, starting from the problem of the
representation of the physical world and from the idea that representing
implies transforming, we see tranformations as a tool for mathematization,
and then we study them in order to give a general frame of reference for
geometry, and we explain the introduction of the concept of invariant as a key
element for organizing it.
MALARA N.A., IADEROSA R.
Theory and Practice: a case of fruitful relationship for the Renewal of the
Teaching and Learning of Algebra', in JAQUET F. (ed.) Relationship between
Classroom Practice and Research in Mathematics Education, Proc CIEAEM
50 (Neuchatel, Switzerland, 1998)1999, 38-54
school level: m; mathematical subject: al educational area: cr.
The paper, which is a wide study at various levels, is divided into three parts.
In the first part, after a presentation on many authors' viewpoints on the effect
of theoretical studies over the teaching praxis, we describe the guidelines of
the Italian research model for innovation, by explaining its genesis and its
more recent evolutions, with particular attention to the role of the teacherresearcher. The second part is specifically devoted to a long term innovative
project for the teaching/learning of algebra and it is focused on the teachers'
role. We specify the theoretical framework of this project, the problems
concerning its enacting as far as teacher's beliefs go, the typology and strategy
of the interventiond carried out with and for the teachers in order to stimulate
and harmonize them with the main goals to be pursued on planning such a
research. We show examples of their contribution to the development of the
project and of the effects on the pupils' activities. The third part contains a
169
reflection on the problems which arose in enacting the research according to
this model, both from the teachers' point of view and from that of the research
director. In particular, initially we concentrate on some problems teachers
have on tackling these research questions with the double role of teacher and
researcher, but wed discuss also the incidence of their beliefs onto the
development of the research and of the influences of the environment. Then
we address the more general problem of the social impact of such kind of
research, also with reference to the current Italiam system for teachers'
training.
As a background to this research there are the following papers:
MALARA N.A., 1996, Algebraic thinking: how is it possible to promote it
already in middle school by limiting its difficulties? Educazione Matematica,
1996, anno XVII, serie V, vol. 1, 80-9, in which we suggest an approach to
algebra as a language. This means we combine the study of: grammar and
syntax (in our case analysis of terms, signs, writing conventions to create
expressions, rules of transformation), translation from one language to the
other (in our case, reading-interpreting formulas in the ordinary language, as
well as expressing sentences of the ordinary language through formulas),
expression in the new language (in our case, argumenting and demonstrating
through algebraic transformations).
MALARA N.A., GHERPELLI L. 1997, Argumentation and proof in
Arithmetics: some results of a long lasting research, proc First Mediterranean
Conference on Mathematics Education , 139-148, a research based on the
hypothesis that an early and substanciated approach to the algebraic language,
giving space to reflection and to the meanings the letters convey, may allow
the pupils to overcome the usual difficulties they have on learning algebra,
and in particular it may lead them to use it autonomously ad with awareness
in demonstration activities.
For a deeper analysis of the curricular aspects concerning the project and
examples of the activities carried out, see MALARA N.A., 1999, Un projecto
de approximación al piensamento algebraico: experiencias, resultadados,
problemas, Revista EMA, vol. 5, n.1, 3-28.
MALARA N.A., RICO L. (eds),
First Italian-Spanish Research Symposium in Mathematics Education, 1994,
AGUM, Modena
school level: -;
mathematical subject: m; educational area: cr, tt, e.
This book contains the contributions to the 1st Italian-Spanish Symposium,
held in Modena in February '94, where the subjects of discussion were: i)
innovation in methodology and curriculum for mathematics (with
presentations by Barra, Coriat, Lanciano, Jimenez, Ortiz, Malara, Scarafioti &
170
Giannetti); teaching/learning problems (with presentations by Boero &
Ferrero, Azcarate & Deulofeu, Chiappini & Lemut, Luengo, D’Amore &
Sandri, Rico & Castro, Gallo, Casro & Alii); contributions to didactics of
mathematics from history and epistemology of mathematics (with
presentations by Arzarello, Gonzales, Bartolini Bussi & pergola, Sierra,
Speranza, Puig); teachers' training in mathematics (with presentations by
Ferrari, Llinares, Furinghetti, Sanchez, Pesci & Reggiani, Rico)
The contribution by Malara, "Didactical Innovation in Geometry for pupils
aged 11-14" (pp. 59-66), aims at giving an overview of the teaching of
geometry in Italian middle school. Initially it outlines the teaching tradition,
on what the current syllabuses contain about this issue, and to the actual
situation in school. Then it exposes the geometry topics for this school level
that were discussed with the teachers. In particular it focuses on the passage
from space to plane and on geometrical transformations. Finally it briefly
describes the most meaningful research our group has lead on these issues.
MARCHINI, C.
La deduzione: esperienze didattiche (Deduction: some teaching experiences),
Ciarrapico L. Mundici D. (a cura di), L'insegnamento della Logica, 1995,
Ministero Pubblica Istruzione e AILA, 159 - 175.
school level: b, t; mathematical subject: al, l
;
educational area: p,
tr, tt.
In 1994 the Italian Public Education Board projected a national in-service
teacher training course in Logic. The first phase was devoted to theoretical
courses. In one of them (Marchini, C.: 1995b, ’Schemi di deduzione’
(Deduction schemes), Ciarrapico L. Mundici D. (a cura di), L'insegnamento
della Logica, Ministero Pubblica Istruzione e AILA, 107 - 132) I present
standard tools used in formal first order mathematical Logic, such as Hilbert's
style deductive systems, tableaux (or refutation trees), natural deduction
system, with examples from school textbooks. In the second phase that take
place some months later, teachers have presented and commented some
activities carried out in their own classroom as an application of the
arguments learned in the first phase. The teachers point out the difficulties of
the organisation of activities on deduction. There are problems about
assessment; different levels of understanding of students prevent a satisfying
activity on deduction. Nevertheless some examples of syntactic logical
aspects have been introduced: analysis of algebraic calculus, games with
playing-cards, logical games. Some attention is paid in the difference between
proof and argumentation. teachers used interrupted proofs, syllogisms,
analysis of textbooks.
171
MARCHINI, C.
Conflitti tra sintassi e semantica nella trattazione delle funzioni, (Semanticsyntax conflicts in the treatment of functions) D'Amore B., Pellegrino C. (a
cura di) Convegno per i sessantacinque anni di Francesco Speranza,
Pitagora, Bologna, 1997, 94 - 98.
school level: b, t; mathematical subject: al, c, l; educational area: e, tr, tt.
Functions are presented in many different ways. In some cases rules or laws
are used, in other cases formulas present algorithms for the calculus of values,
but these approaches use semantic arguments and/or syntactic aspects. The
determination of the domain of the same function can change if we adopt a
point of view or another one. This analysis and also logical aspects used for
the management of free and bounded occurrences of variables are often
neglected. The same argument has been studied in more detailed way in
(Marchini, C.: 1998, ‘Analisi logica della funzione’ (Logical analysis of the
function), Gallo E., Giacardi L., Roero C.S. Eds. Conferenze e Seminari
Associazione Subalpina Mathesis 1997 - 1998, 137 - 157.)
MARCHINI, C.
Il problema dell'area (The problem of the area), L'Educazione Matematica,
1999, Anno XX, Serie VI, Vol 1, 27 - 48
school level: e, m, b;
mathematical subject: g, l
educational
area: cr, d, e, mr, tr.
In the Italian curriculum area is a concept that has to be introduced in the first
two years of elementary school (6 - 7). The paper shows that some aspects of
the concept are suitable for the age indicated in the curriculum, but other are
very profound and must be introduced later. So it is possible to present an
intuitive valuation of the extension, using the psychological principle of
quantity conservation. The development of a measure concept appropriate for
a specific problem is more difficult, since different measures are used in
Mathematics and the same well-known concept of (additive) measure is
inadequate to express the area. The paper analyses some practical methods
used in classrooms: "geopiano", division into squares, Monte Carlo methods,
theoretical methods such as Cavalieri's principle. The poor analogy between
measure of length and measure of surface is emphasized. The paper treats also
some logical aspects connected with substitutions and geometrical formulas.
An intrinsic determination of the area of rectangle using paper folding based
on Euclid's algorithm is shown.
172
MARIOTTI M. A., BARTOLINI BUSSI M. G., BOERO P., FERRI F., GARUTI R.
Approaching Geometry Theorems in Contexts: From History and
Epistemology to Cognition, in Proc. PME 21, Lahti, Finland, 1997, vol. 1,
180-195.
school level: e, m, b;
mathematical subject: g; educational area: p.
This paper is related to the presentation of a forum in PME 21st. It presents
the common theoretical framework and the main findings of three long term
research studies which has been carried out over the last five years by teams
in Genoa, Modena and Pisa. These studies have involved students at different
ages (from grade 5 to grade 10) and different fields of experience, namely the
representation of the visible world by means of geometrical perspective;
sunshadows; geometrical constructions in Cabri environment. An historic epistemological analysis of mathematical theorems as units of statement,
proof and theory, where the conditional form of the statement plays a major
role, is introduced. To approach geometry theorems in this sense, the features
of the field of experience and of the teacher's role in classroom interaction are
analyzed for each teaching experiment. The functions of dynamic exploration
to generate the conditional form of theorems on the one side and the proving
process on the other side are discussed.
Other references:
BARTOLINI BUSSI M. G., BERGAMINI B.,1997, The theorems of Sun: A
Teaching Experiment on Conjecturing and Proving in the 8th Grade, in
Boavida A. M. & al. (eds.), Aprendizagens em Matemática, 21-42, Sociedade
Portuguesa de Ciências da Educaÿão.
MEDICI D., VIGHI P.
Una storia ... improbabile. Introduzione alla probabilità nella Scuola
Elementare (‘A story? An improbable introduction to probability in Primary
School’), L'Educazione Matematica, 1996, Anno XVII, Serie V, Vol.1, n. 2,
58-79.
school level: e;
mathematical subject: p; educational area: pcb.
At first we present a tale, which has been written for an initial approach to
probability in the primary school and in which we have inserted some
questions related to the idea of random events, to equity in games and to
simple comparisons between probabilities. The answers given by the children
enable us to know and analyse their beliefs, their capacity for coping with
situations of uncertainty and for making predictions. Then we continue with
an activity of "dramatisation" in which the pupil, taking the part of the
protagonists in the fable, repeats their experiences, particularly those
connected with uncertain situations. This activity makes the pupil reflect upon
173
his/her answers in a more detached way: in fact the fable has the virtue of
involving, but also the drawback of conditioning the answers a little. The
work has been experimented in class with pupils between the ages of 7 and
11. The reading of the fable text has been interrupted after the presentation of
each question so as to give the children the possibility of reflecting and giving
their answers on purpose-made cards; finally we have held a collective
discussion on the work done, revising the answers written on the cards and
taking into consideration what had been experimentally observed.
MEDICI D., RINALDI M.G., VIGHI P.
Le frecce – Elaborazione ed analisi di alcune schede didattiche (The arrows –
Elaboration and analysis of some didactic exercises), La Matematica e la sua
didattica, 1996, vol. 10, 96-111
school level: e;
mathematical subject: l; educational area: d, mr.
This paper presents an investigation of pupils’ abilities and difficulties on the
use of 'arrow' notation in primary school. The research develops in two
directions: the preparation of some exercises, concerning the arrow
rapresentation of strict order relations, and the study of pupils’ behaviour in
confronting exercises of “direct” type (to draw arrows) or “inverse” type ( to
read and to interpret a graph) In regard to the first one, this paper documents
the successive new elaboration of exercises after experimentations in the
class, necessary to obtain, clear and significant reponses. In regard to the
second one, contrary to all expectation, the results were better in the “inverse”
exercises, indipendentely of pupils’ age. The experimentation involved a
sample of about 200 students.
MENGHINI, M.
The Form in Algebra: Reflecting with Peacock, on Upper Secondary School
Teaching, for the learning of mathematics, 1994, 14, 3, 9-14.
school level: b, t; mathematical subject: hs, al; educational area: e, mr.
After the first two years of upper secondary school the aim of learning algebra
is not to be able to 'do' algebraic calculations of ever increasing difficulty, but
to be able to apply with certainty what has been learnt so far. As part of
educational research then we must ask ourselves what kind of 'control' of
algebraic operations is acquired by the students: checking by substitution, the
recognition of known 'forms', or mastering and recognising the rules of
syntax. The author prefers to get the students into the habit of using the last
two methods of control, i.e. she thinks that it is better, at a certain level, to
underline explicitly the 'leap' from arithmetical algebra to symbolic algebra,
174
giving a major attention to 'form'. To do this she takes a step back into the
past, to the time when the basis for modern algebra was being laid and George
Peacock underlined the two separate aspects of algebra, distinguishing
between independent science on the one hand and 'instrumental' science for
discovery and investigation on the other.
MENGHINI, M.
The Euclidean method in geometry teaching, in Jahnke H.N., Knoche N. &
Otte K. (Hrsg.), History of Mathematics and Education: Ideas and
Experiences, 1994, Vanderhoeck & Ruprecht, Göttingen, 195-212.
school level: b, t: mathematical subject: hs, g;
educational area: cr, p.
The problem of geometry teaching is an age-old one in the history of teaching
and mathematical syllabuses. Even today, in Italy, there are arguments about
the choice between 'traditional euclidean geometry' and so-called
'transformation geometry'.
In Italy, the 'traditional route' is deductive and synthetic teaching of geometry
via axioms and theorems, as done in Euclid's Elements. The tradition begins
when, in 1867, Euclid's Elements were introduced as a textbook. The aim of
this introduction was to improve Italian secondary schooling: geometry, and
thus the Euclidean approach, was seen as 'mental gymnastics'. This reform of
geometry continued, with few changes, until the early 20th century, with
permanent effects on Italian teaching. This reform involves discussion about
'rigid body motions' and about the 'purity' of geometry.
MENGHINI, M.
Klein, Enriques, Libois: variazioni sul concetto di invariante (Klein, Enriques,
Libois: variations on the concept of invariant), I e II parte, L'Educazione
Matematica, 1999, n. 2, 100-109 and n. 3 159-180.
school level: b, t; mathematical subject: hs, g;
educational area: cr, e.
The Erlangen Program has represented, and still represents, the banner under
which one would operate when introducing the new 'transformation geometry'
into the school curriculum. In the 60's and 70's many of those who proposed a
renewal in the teaching of mathematics mantained that the conception of
geometry inspired by Felix Klein's Program no longer allowed a traditional
treatment of Euclidean geometry. Even today, in Italy, the new proposals of
Programs for the triennium (grades 11-13) make specific reference to that
idea.
175
But what real part did Felix Klein's Program have in the proposals to modify
the school curriculum in geometry? The purpose of the paper is to help clarify
into which didactic and scientific view the teacher who inserts transformation
geometry in his/her curriculum places himself/herself today.
In Italy, after the Euroipean reforms of the '50s, the Erlangen program was
eventually accepted, but on the basis of a re-working due to Enriques and his
student P.Libois.
MORELLI, A., TORTORA, R. et al.
Indagine sulla conoscenza e le competenze al passaggio dalla scuola
elementare alla media. Proposte di interventi (Investigation on knowledge and
competence in the transition from primary to middle school. Some Teaching
proposals) - in Numeri e proprietà, atti del I Internuclei Scuola dell'Obbligo,
1994, Univ. di Parma, 87-92.
school level: e, m; mathematical subject: ar; educational area: cr, tcb, va.
Understanding the concept of number, using, operating, comparing decimal
numbers, fractions and relative numbers are among the objectives to be
reached at the end of Italian Primary School. But the opinions about the
relative importance of each ability and the perceptions of the actual
achievement of all these objectives in our schools vary considerably from
one teacher to another, in particular from those operating in the Elementary
School and those in Secondary one. This paper reports an analysis of this
problem. The steps of this investigation are: a questionnaire devised for
teachers, the collection and the elaboration of their answers, a test proposed to
10-years-old students and the comparison of the valuations of the works of
the pupils made by two groups of teachers of the two levels. A proposal will
follow in order to weaken the gap between teaching and evaluating methods
in Elementary and Secondary School, and to facilitate the understanding of
arithmetic.
PAOLA, D.
La multimedialità - Esperienze - Critiche (Multimediality - experiments criticisms), L’Insegnamento della Matematica e delle Scienze Integrate, 1997,
vol. 20A-B, 712-746.
school level: m, b, t;
mathematical subject: m; educational area: cm.
The article is structured into three parts.
1. some general considerations about:
- the meaning of terms like multimedia, hypermedia, intermedia
- the feature of multimedia, hypermedia and intermedia
176
- their cultural influence
- implications for education, in particular for mathematics education.
2. Analysis of some didactic experiments, useful to exemplify the general
considerations of part 1.
3. Description of a teaching experiment in which students aged 15/16 were
engaged into the construction of a hypermedia with the aim of introducing
the concept of equation to students aged 12/13.
PAOLA, D.
Il problema delle parti: Prassi didattica e storia della matematica (The
partition problem: Didactic practice and the history of mathematics),
Didattica delle scienze e informatica nella scuola, 1998, XXXIV, n.198, 3136.
school level: b, t; mathematical subject: hs;
educational
area:
pcb.
This study concerns a teaching experiment carried out with students aged 15.
The classical problem of probabilities known as “the partition problem” was
presented to students. The aim was to introduce students to the concept of
“fair play”. The focus of the paper is on the solving strategies applied by the
groups of students. It was observed that some of these strategies are the same
applied by mathematicians of the past. This suggests some interesting
observations about the use of history in classroom.
PAOLA, D.
Communication et collaboration entre practiciens et chercheurs: étude d'un
cas (Communication and cooperation between teachers and researchers: a
case-study), Proceedings of CIEAEM 50 (Neuchâtel), 217-221
school level: m, b, t;
mathematical subject: p; educational area: tcb.
In this paper I describe an experience of mathematical discussion, that I
proposed to a teacher of a vocational school. I take into consideration the
behavior of the teacher during the discussion. In particular, I underline the
tendency of the teacher to cut short the students’ discussion, legitimizing the
ideas given by the students too quickly. This did not allow students to
recognize the ideas proposed as voices in the class. The analysis of this
experience is intended to give some contribution to the discussion about the
transferability of the ideas of research into practice.
PELLEGRINO C. et al.
177
Perspective from the point of view of Geometry – Per vedere di là della siepe
che da tanta parte dell'ultimo orizzonte il guardo esclude , Pitagora Editrice,
Bologna, 1999, XIV+116
school level: b, t, u;
mathematical subject: g, hs;
educational area: cm,
mr, v.
Perspective has been an important encounter between arts and mathematics
which has turned out to be quite fruitful. The introduction of perspective into
painting, which was the result of a long process aimed at finding out an
effective way of representing scenes and objects realistically, lead to
Renaissance painting which started in Italy and very soon spread all over
Europe. On the other hand, perspective brought into mathematics the seeds of
a process, which matured at the beginning of XIX century, that led to
projective geometry. After a brief overwiew on the origins of perspective, in
which we illustrate the principle of intersection of the visual pyramid on
which it is based, and starting from this simple principle, this paper gives a
simple system of perspective representation through the “power” of the basic
notions of the double projection of Monge, and then “gets to discover” its
fundamental properties and concepts (such as vanishing points, horizon line,
etc.) thanks to the dynamism of Cabri (software purposedly created for
teaching geometry). This way it is possible to illustrate: (1) the genesis of the
rules that are at the basis of various systems of perspective representation; (2)
the origin of the concepts of improper point and improper straight line which,
together with the operations of projection and section, are at the basis of
projective geometry. The study includes considerations on geometry
(homology and conics) and applications (perspective restitution).
PELLEGRINO C., ARPINATI BAROZZI A.M.
Notations and representations as means to dominate complexity: an example
developed in geometrical realm for and with middle school pupils,
L’Insegnamento della Matematica e delle Scienze Integrate, 1996, vol. 19B,
n. 2, 177-193
school level: m; mathematical subject: g; educational area: mr, p, v.
This study is the ideal sequel of a research (carried out for two years with the
same pupils) concerning an aspect that is usually neglected in the teaching of
geometry in middle school: the connection between space and plane. The
core of such research was to identify the plane nets of polyhedra, ad in
particular of cube (see (1)) and octahedron (see (2)). The activity carried out
allowed the pupils, among other things, to explain, by using the right
tridimensional model, why the number of nets of the cube and that of the
octahedron coincide. In this study, on the other hand, we intended to make the
178
pupils realize how important notations and representations are on simplifying
complex procedures. The goal was achieved by elaborating with the pupils a
way of denoting the vertexes and the edges of cube and octahedron that
allowed to achieve the previous result without using any tridimensional
model.
Related papers are:
(1) ARPINATI BAROZZI A.M., PELLEGRINO C., Alla ricerca di una strategia di
classificazione degli sviluppi piani dei parallelepipedi rettangoli (In search of
a strategy to classify the plane nets of rectangual parallelepipeda) , MD, 1991,
n. 4, 4-11
(2)PELLEGRINO C., ARPINATI BAROZZI A.M., Come allievi di terza media
hanno studiato un collegamento tra gli sviluppi dell'ottaedro e del cubo (How
did third-grade middle-school pupils study a connection between the nets of
cube and octahedron), IMSI, 1993, n. 4, 383-398
PELLEGRINO C., ZAGABRIO M.G.
sInvitation to Geometry with Cabri-géomètre - Working cues for upper
secondary school, IPRASE – TN, Collana Strumenti Didattici, 1996, 124
school level: b, t; mathematical subject: g; educational area: cm.
The use of new programming languages and software allows us to provide a
more dynamic and creative look at mathematics. This, however, doesn't mean
that one should always look for the most updated and powerful software,
because, quite paradoxically, the educational and formative aspects of
matematics are better enhanced by simple and essential software. As to this
issue, the basic version of Cabri is very interesting just because, unlike other
software, it doensn't allow the operator to use control instructions, make
calculations or manipulate formulas.
In order to make this idea more explicit, we carried out a study aimed at
illustrating the didactical potentialities offered by Cabri to the teaching of
geometry in secondary school. From the disciplinary point of view, this study
paid particular attention to the explicitation of the links between synthetic
geometry and geometry of transformation, without neglecting its analytical
aspects. From a merely educational point of view, in order to make the
presentation more interesting and fruitful, we paid particular attention to the
heuristic aspects and to the contents connected to the identification and
refinement of the solutions of the problems suggested. See also
PELLEGRINO C., BAROZZI E., 'Geometrical explorations: Cabri and
isometries', MD, 1997, n. 2, 202-212
school level: b, t; mathematical subject: g; educational area: cm.
PELLEGRINO C., BAROZZI E., 'Geometrical explorations: From where was
this picture shot?', in D'Amore B., Pellegrino C. (eds.), Convegno per i
179
sessantacinque anni di Francesco Speranza, Pitagora, Bologna, 1997, 118123
school level: t, u; mathematical subject: g; educational area: cm.
PELLEGRINO C., BONACINI B, 'Geometrical explorations: Parabolas and
similarities', MD, 1997, n. 1, 69-73
school level: b, t; mathematical subject: g; educational area: cm.
PELLEGRINO C., ZAGABRIO M.G., 'Geometrical explorations: Cabri and
affinities', MD, 1998, n. 4, 458-468
school level: b, t, u;
mathematical subject: g; educational area: cm.
PESCI A.
Tree graphs: visual aids in casual compound events, Proc. PME 18, 29 July-3
August 1994, vol IV, 25-32.
school level: m, b; mathematical subject: p; educational area: d, mr, v.
In this study an analysis of tree graphs as graphical representations which can
work as visual aids in the understanding and in the solution of casual
compound events is proposed.
It is reported how tree graphs can describe, in figural terms, the conceptual
relationships involved and stimulate the use of an adequate calculation
procedure.
It is then examined how tree graphs are used by 13-14 year old students to
solve two problems with different characteristics.
After a prior study of the two problems mentioned above and a description of
the "theoretical background" in the classes where the investigation was
carried out, an analysis of the results achieved is proposed and the most
significant errors are examined.
PESCI A.
Visualization in Mathematics and graphical mediators: an experience with 1112 year old pupils', R. J. Sutherland, J. Mason (Eds.), Exploiting Mental
Imagery with Computers in Mathematics Education, Nato ASI Series F/138,
1995, 34 - 51.
school level: m; mathematical subject: ar; educational area: cr, d, mr, v.
The paper is divided into three main sections. In the first one there is a
collection of contributions from recent literature about the process implied by
terms such as to ‘visualize’, to ‘imagine’, etc reported by researchers in
mathematics education. Some statements on the role of graphical
representations in mathematics activities are also reported. The second section
is a description of a study carried out by our research group in connection
with the theme ‘visualization in mathematics education’. It deals in particular
180
with arithmetical inverse problems, and in general with the concept of inverse
function (with 11-12 year old pupils). The graphical sign used as ‘mediator’ is
the arrow scheme: the hypothesis is that it can be ‘the (concrete) carrier’ for
the related concepts, as specified in detail. The last section contains some
methodological observations derived from the described study which may
have more general value: in other words they may be relevant for other
studies dealing with the analysis of the efficacy of visual aids in mathematics
teaching and learning.
The same work, toghether with further theoretical notes, is also quoted in
Pesci A., 1997, Il ruolo della visualizzazione nella costruzione dei concetti
matematici, Conferenze e seminari 1995-1996, Associazione Subalpina
Mathesis, Seminario di Storia delle Matematiche “Tullio Viola”, E. Gallo, L.
Giacardi, C. S. Roero (Eds.), 160-169.
PESCI A.
Class discussion as an opportunity for proportional reasoning, in Proc. PME
22, Stellenbosch, South Africa, 1998, vol. 3, 343-350.
school level: m; mathematical subject: ar; educational area: cr, d.
The initial phase of the construction of proportional reasoning with students
aged 12 to 13 is described. In the proposed problematic situation, the recourse
to the constance of ratios originates as a strategy necessary to tackle this same
situation and more importantly it clashes with other resolution strategies
which are quite spontaneous but not really suitable.
The fundamental part of the didatic proposal turned out to be the discussion
conducted by the teacher, which gave the students the opportunity to freely
express their agreement or disagreement, even with original argumentation
from the mathematical point of view.
In reference to current constructivist perspectives, the main points of contact
are underlined, with a particular emphasis on social constructivism.
PESCI A.
Mathematical models and hypothetical reasoning when students approach
proportionality, in A. Rogerson (ed.), Proc. of the International Conference
on Mathematics Education into the 21st Century: Societal Challenges, Issues
and Approaches, Cairo, Egypt, 1999, Vol. I, 265-272.
school level: m; mathematical subject: areducational area: cr, m, pcb.
Hypothetical reasoning in mathematical problem solving has been analysed in
some interesting research works, which put in evidence for instance the
181
conditions for the production of this type of reasoning or the different
functions it may have during the process of problem solving.
This contribution has the aims of specifying some mathematical models
proposed by students while they are approaching two proportional problems,
before the subject ‘proportionality’ is treated in class by the teacher; of
showing some examples of students’ hypothetical reasoning which are present
both in their individual protocols and while they are discussing among
themselves with the goal to prove or to refute the models proposed in class as
solution strategies for the given problem situations; of stressing different
functions of hypothetical reasoning during the process of problem solving, in
agreement with some of the typologies described in literature and developing
them further.
The study is based on didactical experiences carried out with 12-13 year old
students about the construction of proportional reasoning.
POLO M.
Le repère cartésien dans les systèmes scolaires français et italien: étude
didactique
et
application
de
méthodes
d’analyse
statistique
multidimensionnelle (Cartesian coordinates in french and italian school
systems: a didactical study with application of multidimensional statistical
analysis). Thèse en Didactique des Mathématiques, University of Rennes I –
I.R.M.A.R., 1997, Rennes.
school level: me, m, b, t, u;
mathematical subject: al, g, hs; educational
area: d, pcb, tr
This study as a whole is part of the broader context of research which takes
into consideration the dialectic contribution of didactical analysis (Theory of
situations and Didactical contract, Brousseau, 1986-1990; Didactical
transposition, Chevallard, 1985) and statistical analysis (the statistical
implication, R. Gras, 1991-1996) in order to identify specific phenomena of
the institution of the School as a system. The system of Cartesian reference,
as taught knowledge, officially present in syllabi throughout the course of
study from middle school to secondary schools in Italy and France, is taken
into consideration. An analysis of textbooks, relative to this period of study,
used in schools in the two countries, identifies the institutional relations of the
teacher and the student to this knowledge. It provides an index of the
reduction of the sense of this knowledge. The analysis of some of the effects
of this reduction on students’ knowledge at university entrance level
completes the study. The theoretical lines of the method, elaborated by R.
Gras and his collaborators, are presented in R. Gras et al., L’implication
statistique. Nouvelle méthode exploratoire des données, published by La
Pensée Sauvage, 1996, Grenoble, and in Polo M., 1995, Traitement de
182
résultats d’un questionnaire portant sur la représentation graphique, Actes du
Colloque de Caen, 27-29 January 1995, Méthodes d’analyse statistique
multidimensionnelle en didactique des mathématiques, pp. 181-198, ARDM,
IRMAR, Rennes.
POLO M.
Il contratto didattico come strumento di lettura della pratica didattica con la
matematica (The 'Didactical contract' as a tool for interpreting tesching
practice in mathematics), L’Educazione Matematica, 1999, vol. 1, no. 1,
February 1999, 4-15
school level: e, m, b;
mathematical subject: m; educational area: tr, tt.
By using the theoretical concept of the Didactical contract relative to
knowledge, briefly presented in paragraph 2 according to the theoretical lines
of the studies by Brousseau 1986-1990, a specific point of view on the social
interactions regulating “life in the classroom” is described. The aim of the
study is to provide a key to reading didactical practice in order to reach an
analysis of the nature of the answers given by the students. In particular, the
different role of the error in learning processes, depending on the knowledge
acquired or knowledge in the phase of construction, is analysed and
exemplified. The role of the teacher depending on this diversification is
briefly described.
REGGIANI M.
Analisi di difficoltà legate all'uso di convenzioni nel linguaggio aritmeticoalgebrico' (' Analysis of difficulties connected to the use of conventions in the
language of arithmetic and algebra'). Atti I Internuclei Scuola dell'obbligo,
Salsomaggiore Terme (PR), 1994, 14-16/4/1994, 61-66
school level: m; mathematical subject: ar, al; educational area: d.
It is thought that the difficulties the students have in using algebraic
language, in the first years of lower secondary school, originate from
obstacles of a prealgebraic nature. The attention of our research group has
been focused for some time on the determination of nodal moments in the
study of arithmetc and in the arithmetic-algebra passage, in which it is
possible to find the origins of misunderstandings, or on the contrary an
introduction that leads to a good use of algebraic formalism. One of these in
our opinion, is the awareness of the conventions used in the fields of
arithmetic and algebra. Here we wish to present some observations on the
conventions, with particular reference to relative numbers. Such observations
come from students errors noted during an activity not directed towards this
183
topic and from the examination of the results of a test created for the purpose.
There are also some of the students' verbalizations concerning the topic in
question. The study was carried out in the third year of lower secondary
school (14 year old students).
REGGIANI M.
Insegnare a programmare nella scuola media inferiore: obiettivi, risultati,
difficoltà, riflessioni ('The teaching of a programming language in lower
secondary schools: aims, results, difficulties, observations'), L'Insegnamento
della matematica e delle scienze integrate, 1994, vol. 17 B, n.1, pagg.65-91
school level: m; mathematical subject: m; educational area: cm.
The teaching of a programming language in lower secondary schools and its
use in close connection to the teaching of Mathematics can be interesting and
productive for both teachers and students.
However it is necessary for the teacher to have clear objectives and to
understand the limits of this work and the difficulties that are involved in the
use of a formal language and in particular that of programming and its
structures. This article proposes to present some results and to supply some
opportunities for reflection based on a didactic experience and on research
studies carried out on learning processes connnected to it.
REGGIANI M.
Generalization as a basis for algebraic thinking: observations with 11-12 year
old pupils Proc. PME 18, Lisbona, Portugal, 1994, vol.4, 97-104
school level: m; mathematical subject: al; educational area: tr.
Generalization, intended as the ability to pass from the particular to the
general or also the ability to see the general in the particular, represents a
fundamental element of algebraic thinking that would otherwise be simply
working with symbols. Moreover it is a common experience that often the
pure technical aspect prevails in didactic practice and the operative ability
sometimes hides the lack of understanding of the general significance of that
which is being done. The bases of algebraic thinking are laid when the
properties of the operations between numbers are learnt and one starts to work
with symbols in various contexts (arithmetical, geometrical, data processing),
but the acquisitions in this field must be considered as a gradual achievement
and must be continually consolidated. Our research study proposes to
determine the moments of origin of some of the difficulties that consequently
cause errors and misunderstandings in working algebraically. This report
refers to the level of generalization in 11-12 year old pupils, showing how the
184
ability to grasp the generality of the result may be apparent and trying to
understand when it corresponds to real awareness of general relationships
REGGIANI M.
Il ruolo dell’argomentazione nell’approccio all’algebra (The role of
argumentation in the approach to algebra), in Grugnetti L. et al. (eds)., Atti
XV Internuclei Scuola Media “Argomentare e dimostrare nella scuola
media”, 1996, 24-31
school level: m; mathematical subject: al; educational area: d.
In the frame of a didactic approach to algebra with students aged 11 to 14,
there are many situations where argumentation plays an important role. It
happens, for instance, when it is necessary to formulate conjectures, to
generalize solutions in numerical problems, to understand whether an
algebrical manipulation is correct or not, to make some examples or to use
some graphical representations as non verbal argumentations.
In this report some didactical situations are presented and discussed.
REGGIANI M.
Continuità nella costruzione del pensiero algebrico ('Continuity in the
building of algebraic thinking'). Atti convegno UMI- CIIM, (Campobasso,
1996), Suppl. Notiziario UMI n.7, 1997, 35-48
school level: m, b; mathematical subject: al; educational area: cr.
The continuity in the teaching of arithmetics and algebra in the passage from
a school level to another, is seen in this paper as an instrument aiming at the
building of algebraic thinking. By this term we mean the ability to use algebra
as a means for representing a situation and as an instrument for solving a
problem. Starting from the analysis of the official national curriculum and
referring to many studies about this issue, the paper analyzes some moments
that are central in this proposal for a way from the elementary school to the
two first years of high school.
It’s very important to see algebra as a language and not only as a matter of
studying and to propose it as a tool for codification and generalization in
different situations.
REGGIANI M, TOMASSINI F.
Un’attività didattica sull’equiestensione ('A didactic activity about the
concept of figures with the same area'), ) in Grugnetti L & Gregori S. (eds.)
185
Dallo Spazio del bambino agli spazi della geometria, Atti del 2° Internuclei
Scuola dell'Obbligo, 1997, Parma, 83-88
school level: m; mathematical subject: g; educational area: cr.
This paper describes a teaching project which aims at giving an understanding
of the concept of figures with the same area through the division of figures
into congruent polygons, using the tangram geometric puzzle as a teaching
aid.
After initial play activity to familiarize pupils with this aid, more specific
activities are introduced.
A fundamental role is played by making, examining and comparing the
different figures produced during group work and by discussing the pupils’
explanations of their work.
This project develops an understanding of the formulae for areas of the
principal polygons and an approach to problem solving using demonstration
and discussion.
ROCCO M.
La misura in Cabri-Géomètre: esempi di risposte dello strumento e
implicazioni didattiche per la scuola media' ('Measuring with CabriGéomètre: examples of feedback by the tool and educational implications for
middle school') in Boeri P. (ed.) Fare geometria con Cabri (Doing Geometry
with Cabri), Centro Ricerche Didattiche U.Morin, 1996, 75-83
school level: m; mathematical subject: g; educational area: cm.
The paper deals with the use of the measuring opportunities of CabriGéomètre at the age of 12 and with the warnings related to approximation
effects. Some examples of control by means of mental computation on the
incorrect screen visualization are given. Ordinary and screen geometry are
compared with some justification of the effects observed. A quantitative
evaluation of measurement errors related to expected measures is given. The
experience is related to mathematics curriculum. Some activities related to the
discovery of theorems are presented.
SCALI E.
Choix des taches et organisation des interactions dans la classe pour
l'appropriation des signes de la géométrie dans les activités de modélisation
('Choice of tasks and organization of the classroom interactions for the
mastery of geometry signs within modeling activities'), Proc. of CIEAEM-49,
Setubal, Portugal, 1998, 186-194
school level: e;
mathematical subject: g; educational area: cr.
186
This contribution concerns the appropriation of geometry signs by primary
school students as tools in the activitiens of geometrical modelling of natural
phenomena (in our case, sun shadows) and the role of the teacher in it. I will
distinguish between direct mediation and different kinds of indirect
mediation. Particularly, I will stress the importance of indirect mediation
exerced through the choice of suitable tasks engaging students in the
prevision and interpretation of phenomena.
SCALI E.
Intégration entre recherche et pratique professionnelle des enseignants: étude
d'un cas (Integration between research and teachers' professional practice: a
case-study), Proc. of CIEAEM-50, (Neuchatel, Swizerland, 1998) 1999, 198202
school level: e;
mathematical subject: m; educational area: tt, tr.
This contribution concerns an example of progressive integration between
research and classroom practice, strictly related to the evolution of the
activities and perspectives of the Genoa Group for mathematics education in
compulsory school over the last twenty years. The object of the reported
example is the construction of the "value" meaning of the number concept
(i.e. the meaning which our current system of writing numbers depends on) at
the beginning of primary school.
SCIMEMI B.
Studio delle similitudini piane con l'ausilio del calcolatore ('Study of plane
similarities with the aid of computer'), Atti del XVI Convegno
sull'insegnamento della matematica, Latina (1994) , in Notiziario 'U.M.I.
1995.
school level: b, t; mathematical subject: g; educational area: cm, v.
For each family of plane isometries (translations, rotations, reflections etc.)
and similarities (e.g. homotheties) we describe specific constructions by ruler
and compass and give detailed instructions to achieve them by CABRI, a
popular software for geometric design by standard PC. Several products of
specific transformations are also studied in detail, thus giving an insight into
the group properties of geometrical transformations.
187
SCIMEMI, B.
Come si vedono i lati di un triangolo - Una collaborazione tra geometrie
vecchie e nuove, (How the sides of a triangle are viewed - An example of
cooperation beween old and new geometries), Archimede, 46, 4(1996), 174181
school level: t, u; mathematical subject: g; educational area: v.
In the Euclidean plane we study the locus of points which “see” two sides of a
triangle under the same angle. This locus comes out to be a cubic curve
(obviously containing the Fermat’s points), which is then studied by both
analytic and synthetic methods. This suggests a number of more general
considerations, e.g. the definition of an involutory plane transformation under
which this curve is invariant .
SCIMEMI B.
Riscoprendo la geometria del triangolo (Re-discovering the geometry of
triangle) - Quaderni del Ministero della Pubblica Istruzione, quad.19/2 Seminario formazione docenti 95-96: L'insegnamento della geometria, Lucca,
1997
school level: b, t; mathematical subject: g; educational area: p.
This is a collection of classical results on the geometry of the triangle
(reflections on mirrors and billiards, minimum inscribed triangle, Fermat’s
point, Euler’s line, 9-point circle etc.) which can be used by high-school
teachers who want to experiment how the use of geometric transformations
(isometries and simililarities of the Euclidean plane) can conveniently replace
more traditional approaches.
SCIMEMI B.
Contrappunto musicale e trasformazioni geometriche' (Musical counterpoint
and geometrical transformations) - Atti del Convegno "Matematica e cultura",
Venezia, 1997, suppl. a Lettera Pristem 27-28, 1997, p.87-9
school level: b, t; mathematical subject: g; educational area: p.
Symmetries have been used by architects and painters in order to enrich the
value of their works. They have also been used in music, but it is not always
trivial to perceive and appreciate their presence. This paper is an introduction
for non-specialists to some typical counterpoint rules in traditional Baroque
music. By using the language of geometric transformations, we consider
several examples of canons, mostly taken from Bach's famous works. All
examples are effectively represented by proper graphs. This article is the
188
report of a lecture which was actually illustrated by colour-transparencies and
simultaneous musical performances.
SPAGNOLO F.
Epistemological obstacles: The Eudoxe-Archimede Postulate, in A
multidimensional approach to learning in Mathematics and Sciences,
Intercollege press, Cyprus, Nicosia 1999.
school level: u; mathematical subject: c educational area: e, pbc, tr.
The article presents a research on epistemological obstacles in mathematics.
The hypothesis that hte obstacles derive from the Postulate and from
languages is here put forward. The representation of the epistemological
obstacles proposed by Duroux-Brousseau, starting from Bachelard's work,
allows one to recognise, at most, whether a piece of knowledge is an obstacle.
It does not provide any means, before hand, for the research of the obstacles.
The attempt to define a standard for the definition of the epistemological
obstacles, which is neither historic nor didactic, has led us to adopt a semiotic
approach to mathematics. With reference to the theory of situations and
through a semiotic approach to mathematics, we have shown that an obstacle
is connected to an important character of the language: The obstacles must,
first of all, be looked for in the changes of the Postulate. These obstacles, too
universally accepted, by too long as being evident and indispensable. The
Eudoxe-Archimede Postulate is a piece of knowledge which presents an
epistemological obstacle to preliminary introduction of hyper-real numbers
and could be an obstacle to the comprehension of non-standard analysis.
TIZZANI P., BOERO P.
La chute des corps de Aristote à Galilée: voix de l'histoire et echos dans la
classe pour l'approche au savoir théorique ('The fall of bodies from Aristotle
to Galilei: the voice of history and echoes in the classroom for the approach to
theoretical knowledge' , Proc. of CIEAEM-49, (Setubal, Portugal, 1997) 1998,
369-376
school level: m; mathematical subject: hs;
educational area: p, pcb.
This workshop concerns the analysis and discussion of some phases of a
teaching experiment regarding a new didactical method, the "Voices and
echoes game". The experiment was perfomed in some classes whose teachers
are members of the Genoa Research Group (level: comprehensive school,
grade VIII). This method was conceived in order to allow all students to
productively approach theoretical knowledge (in the case considered in this
189
workshop, students approached Galilei's and Aristotle's theories about fall of
bodies and related problems of mathematical modelling).
TONELLI M., ZAN R.
Il ruolo dei comportamenti metacognitivi nella risoluzione di problemi.,
L’Insegnamento della Matematica e delle Scienze Integrate, 1995, vol.18A,
n.1.
school level: u; mathematical subject: m; educational area: m.
In this paper the role of metacognitive processes in problem solving is
emphasized. After a general introduction the authors specifically deal with the
concept of strategic decisions and Schoenfeld’s related model. The protocol
of a first year student in Mathematics is presented, concerning a problem in
geometry. In order to analyze the student’s metacognitive processes the
related model of Schoenfeld is discussed and a different model is proposed.
VACCARO,V.
Un mondo fantastico per le frazioni (A wonderful world for fractions),
L’Insegnamento della Matematica e delle Scienze Integrate, 1998, 21A, 321352.
school level: e;
mathematical subject: ar; educational area: d, mr.
This paper contains a proposal concerning the teaching of fractions. It
consists of two parts. In the first one, we briefly resume the wide debate about
the difficulties which underly the learning of fractions, then we specify the
characteristics of our proposal, report the results of the first teaching
experiments and, after some changes made in the proposal, the final results.
The second part contains the details of the didactical proposal, namely a story
equipped with a series of operative cards. Children discover improper
fractions and equivalent fractions in a "fantastic" way, if the story is used as a
first approach. If children already know these notions, they discover them
again but from another point of view and with some improvements. The
protagonists of the story have to solve a difficult problem, if they want to save
their heads! Children, working as they were the protagonists (and with the aid
of an other character who, fortunately, knows mathematics), become aware of
the necessity of knowing something about fractions in order to solve the
problem.
VENÈ, M., MELEJ, A., DEL FRATE M.G., FERRARIS M.L., MAFFINI A.,
CERVI C.:
190
Dai grafici al concetto di limite tramite l'analisi non standard (From graphs to
the concept of limit through non standard analysis), 1994, Atti del IV Incontro
Nuclei di Ricerca Didattica in Matematica, Siena.
school level: t, u; mathematical subject: c; educational area: cr.
The article deals with an experiment about non-standard Analysis, which was
intended as a preliminary introduction to concepts and methods of classical
analysis. The experiment was carried out, with positive results, both in senior
high school and in preparatory courses for first year of university.
VIGHI P.
Dalle opere di Escher alle trasformazioni geometriche (From Escher's works
to geometric transformations), Didattica, 1994, Anno III, n.1, 1996, 75-85.
school level: b; mathematical subject: g; educational area: cr.
We present the main phases of a teaching module (for students between the
ages of 14 and 16) focused on the introduction of basic geometrical plane
transformations, starting from the drawing of M.C. Escher. The module is
divided into teaching units, which are described not only as regards the
contents and the ways of presenting them, but also as regards the students'
reactions, their discoveries and difficulties. The experimental procedure has
followed this pattern: the students are placed in an interactive situation, they
are shown the engravings of Escher, more and more complex, then they are
helped to observe and identify the type of transformation and its properties;
finally the work done is summarised, the statements and theorems are
formulated. The module, which involves the transition from drawing to
mathematization and vice versa, yields the idea of mathematics as an
instrument through which you can organise and rationalise thought.
VIGHI P., SPERANZA F.
Spazio dell’arte, spazio della matematica ('Space of art, space of
mathematics'), in Cerasoli M., Freguglia P., Maturo A. (Eds.), Atti Convegno
Nazionale Mathesis, Arte e Matematica: un sorprendente binomio, Vasto
(CH) 14-16 marzo 1997, 1997, 289-295.
school level: -;
mathematical subject: g;
educational area: e.
It is a work on the theme of "mathematics and culture". The concept of
"space" is discussed from the point of view of mathematics (geometry), art
(particularly the Italian art of the 14th century) and philosophy. We stress the
importance of dealing with the subject as it were a single theme, but treating it
from different points of view; we give examples, we draw parallels, we
191
emphasise analogies and differences, we stress how art has sometimes
anticipated mathematics or vice versa: we conclude with considerations
pertaining the teaching field and with a plea for giving back to mathematics
its educational and formative role.
ZAN R.
Problemi e convinzioni. Pitagora Editrice Bologna, 1998.
school level: e, m; mathematical subject: m; educational area: m, pcb.
This book contains six contributions, previously published in journals or in
proceedings of conferences, about the role of metacognition and affect in
problem solving. More precisely the first chapter, in collaboration with
Efraim Fischbein, deals with the ability of selecting the relevant data to solve
a standard arithmetical problem. In the second chapter the results of an
inquiry are presented, which involved 750 pupils of elementary schools,
aimed at recognizing and comparing the pupils’ models of ‘mathematical
school problem’ and of ‘real problem’. The third chapter suggests, through
the results of a study that involved 300 pupils aged between 8 and 9, the
relevance of a particular variable (called the ‘level’ of a problem), connected
with motivational aspects, in the context of a word problem. The fifth chapter
deals with the role of beliefs in mathematical problem solving. Finally, the
fourth and sixth chapters present an approach to learning difficulties in
mathematics, which emphasizes the role of metacognition, beliefs and affect
in mathematical problem solving, and, more generally, in mathematics
learning.
ZAN R.
A metacognitive intervention in Mathematics at University level,
International Journal Mathematics Education Science and Technologies,
2000, vol.31, n.1, 143-150
school level: u
mathematical subject: c, g
educational area: m, pcb.
In this paper the main results of an instructional study are presented. The
study was aimed at improving the performance in mathematics of a group of
university students of biology who repeatedly failed the final examination of a
compulsory course in mathematics. The main difficulties of these students
seemed to be metacognitive and affective in nature. Therefore the training
worked on metacognitive and affective features: knowledge about cognition,
monitoring, beliefs, emotions and attitudes. The intervention was successful:
at the end of the course all students passed the examination that they had
192
failed so often. The results also suggest that it may be possible (and
necessary) to “teach learning to learn” Mathematics.
ZAN R.
Difficoltà d’apprendimento e problem solving: proposte per un’attività di
recupero.’, L’Insegnamento della Matematica e delle Scienze Integrate, 1996,
vol.19B, n.4.
school level: b; mathematical subject: m; educational area: m, pcb.
In this paper an approach to difficulties in mathematics is proposed, that
involves cognitive, metacognitive and affective variables: in particular
beliefs, emotions and attitudes. It is suggested that a problem solving
instruction can help many students to develop metacognitive abilities, to make
explicit their beliefs and misconceptions, and to become responsible for their
own learning. The kind of approach described is therefore used to interpret
the difficulties of 15 students in a second class of a high school, and to plan
an instructional intervention. The results of this intervention are presented and
discussed.
ZAN R.,
Students’ and teachers’ theories of success in Mathematic’, in G. Philippou
(ed.) Proceedings of MAVI 8, Cyprus, 11-15 1999
school level: b, t; mathematical subject: m; educational area: pcb, tcb.
In the context of Mathematics learning difficulties, beliefs appear to be
important not only because they can inhibit the utilization of knowledge and
then lead to failure, but also because they influence the interpretation of that
failure. Particulary relevant in this sense are the theories of success, i.e. the
beliefs that an individual has about success (in Mathematics). After a brief
theoretical introduction, the paper presents some preliminary research
findings of a study, aimed at investigating 386 high school students’ theories
of success, and at comparing them with 30 high school teachers’ theories of
success.
ZUCCHERI L.
Semitransparent mirrors as tools for geometry teaching', in Inge Schwank
(Ed.), Proceedings of the First Conference of the European Society for
Research in Mathematics Education, 1999, vol. 1, 282-291
school level: e, m; mathematical subject: g; educational area: d
193
The author presents and discusses an experience in Geometry teaching, made
in several years in co-operation with teachers of the Didactic Research Group
of Trieste, starting from 1987. This experience is based on the use of a
didactic tool, which consists of semitransparent mirrors, used in laboratory
activities. This work is arisen at primary school level and was extended
further at middle and secondary school level, realizing also a Mathematics
hands-on exhibition with didactic purposes. This exhibition, named "Oltre lo
specchio" ("Beyond the mirror"), operated in Trieste (Italy) from 1992 to
1997 and was visited by many thousands of people. In this paper the author
describes the used tool and its characteristics from educational point of view;
moreover she illustrates some significant examples of its utilization for
Geometry teaching, among those presented in the exhibition.
Acknowledgements
We wish to tank Leo Rogers for his revising of the translations.
194
KEYS FOR CLASSIFICATION OF ABSTRACTS
In order to give an outline of the orientation of the research we have asked the
authors to classify their papers according to the following streams:
School level
c
e
m
b
t
u
= Infant school
= Elementary school
= Lower secondary school
= Initial two yeras of upper secondary school
= Last three years of upper secondary school
= university level.
Mathematical subject
a
ar
c
g
hs
l
p
s
= Algebra
= Arithmetic
= Calculus
= Geometry
= History of Mathematics
= Logic
= Probability
= Statistic.
Educational area
ap
cm
cr
d
i
m
mr
p
pcb
ps
tcb
tr
tt
v
va
= applications; as = Astronomy
= Computer and Mathematics
= Curriculum Research
= Didactics; e = epistemology
= Image of mathematics
= Metacognition, social and affettive factors
= Models and representations
= Proofs
= Pupils Beliefs and Conceptions
= Problem Solving
= Teachers Beliefs and Conceptions
= Theoretical educational Research
= Teacher Training
= Visualization
= Evaluation
89
ADDRESSES OF THE AUTHORS
ACCASCINA Giuseppe, Dipartimento di Metodi e Modelli Matematici per le
Scienze Applicate, Università di Roma 'La Sapienza', via A.Scarpa 16, 00161
Roma, e-mail: [email protected]
AGLI Francesco Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
AJELLO Maria Elena, c/o Gruppo di Ricerca Insegnamento delle Matematiche,
Dipartimento di Matematica e Applicazioni - Via Archirafi, 34 - 90123
Palermo
ANDRIANI Maria Felicia, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
ANDRIANO Valeria, Dipartimento di Matematica, Università di Torino, 10123
Torino
ARCHETTI A. (c/o Cannizzaro)
ARDIZZONE M.R. (c/o Lanciano)
ARMIENTO S. (c/o Cannizzaro)
ARPINATI BAROZZI A.M., IRRSAE Emilia Romagna, via U. Bassi, 40100
Bologna, e-mail: [email protected]
ARRIGO Gianfranco Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
ARZARELLO Ferdinando, Dipartimento di Matematica, Università di Torino,
10123 Torino, e-mail: [email protected]
AURICCHIO V. (c/o Lemut)
BAGNI, Giorgio Tomaso, Via Venanzio Fortunato 28, I-31100 Treviso, Italy
e-mail: bagni,[email protected]
BALZANO E., Dipartimento di Scienze Fisiche, Università "Federico II", Via
Cintia, Complesso Monte S. Angelo - 80126 Napoli, email
[email protected]
BARDONE Luigi (c/o Pesci)
BARTOLINI BUSSI Mariolina, Dipartimento di Matematica Pura e Applicata
'Giuseppe Vitali', Università di Modena e Reggio Emilia, via G. Campi
213/B, 41100 Modena, e-mail: [email protected]
BASILE E. (c/o Cannizzaro)
BASSO Milena, Via Palù 81, 35017 Piombino Dese (PD), tel. +49-9365351,
fax +49-9365081
BAZZINI Luciana, Dipartimento di Matematica, Università di Torino, 10123
Torino, e-mail: [email protected]
BECCHERE Maria, C.R.S.E.M. c/o Dipartimento di Matematica, viale Merello
92, 09123 Cagliari
BEDULLI Marcella, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
BERNARDI Claudio, Dipartimento di Matematica, Università La Sapienza,
Piazzale A. Moro 2, 00185 Roma, e-mail: [email protected]
BERNESCHI P. (c/o Accascina)
BERTONI Vera., Nucleo di Ricerca Didattica , Dipartimento di Matematica,
Università di Torino, Via Carlo Alberto 10 , 10123 Torino
BOERO Paolo, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, e-mail: [email protected]
BONI Mara (c/o Bartolini Bussi)
BONOTTO Cinzia, Università degli Studi di Padova, Dipartimento di
Matematica Pura e Applicata, via Belzoni 7, 35131 Padova, fax +0498275892, e-mail: [email protected]
BORNORONI S (c/o Accascina)
BOSCO Arturo, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
BOTTINO Rosa Maria, Consiglio Nazionale delle Ricerche, Istituto per la
Matematica Applicata, Via de Marini 6, 16149 Genova, e-mail:
[email protected]
BOVIO Mauro, Gruppo di Ricerca Didattica c/o Dipartimento di Matematica
Università, Via Abbiategrasso 215, 27100 Pavia
CANNIZZARO Lucilla, Dipartimento di Matematica, Università La Sapienza,
Piazzale A. Moro 2, 00185 Roma, e-mail: [email protected]
CAPELLI Laura, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
CAREDDA Carla, Dipartimento di Matematica, viale Merello 92, 09123
Cagliari, e-mail: [email protected]
CARLUCCI Antonella, (c/o Boero)
CASELLA Francesco, I.R.R.S.A.E, Via Traversa, 85100 Potenza
CASSANI Alida Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
CASTAGNOLA Ercole, Via Palazzo 4, 04023 Formia (LT), e-mail:
[email protected]
CASTRO Chiara, Nucleo di Ricerca in
Dipartimento di Matematica, Università di
Donato 5, 40126 Bologna
CHIAPPINI Giampaolo, Consiglio Nazionale
Matematica Applicata, Via de Marini
[email protected]
Didattica della Matematica,
Bologna, piazza di Porta San
delle Ricerche, Istituto per la
6, 16149 Genova, e-mail:
183
CILENTO E. (c/o Lanciano)
CIMADOMO Rosanna, I.R.R.S.A.E, Via Traversa, 85100 Potenza
CROCINI Paola (c/o Cannizzaro)
CROSIA Luigi (c/o Pesci)
CUTUGNO (c/o Bottino)
DALLANOCE Silvia, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
D'AMORE Bruno, Dipartimento di Matematica, Università di Bologna, piazza
di Porta San Donato 5, 40126 Bologna, e-mail: [email protected]
DAPUETO Carlo, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, e-mail: [email protected]
DELEONARDI Claudia Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
DELL'AMURA Mariarosaria, (c/o Morelli)
DELLA ROCCA G. (c/o Accascina)
DE LUCA Giuseppe, Dipartimento di Scienze storiche, linguistiche e
antropologiche - Università della Basilicata, V. Acerenza, 85100 Potenza
DELUCCHI Stefania, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
DEMATTÈ Adriano, via Madonnina 14, 38050 Povo di Trento, e-mail:
[email protected]
DETTORI GIULIANA, Consiglio Nazionale delle Ricerche, Istituto per la
Matematica Applicata, Via de Marini 6, 16149 Genova, e-mail:
[email protected]
DE VITA M. (c/o Accascina)
DI LEONARDO Maria Vittoria, Dipartimento di Matematica e Applicazioni,
Università di Palermo, via Archirafi 34, 90123 Palermo
FASANO Margherita, Dipartimento di Matematica - Università della
Basilicata, V. N. Sauro 85, 85100 Potenza, e-mail: [email protected]
FERRANDO Elisabetta (c/o Boero)
FERRARI Mario, Dipartimento di Matematica – Università, Via Abbiategrasso
215, 27100 Pavia, e-mail: [email protected]
FERRARI Pier Luigi, Dipartimento di Scienze e Tecnologie Avanzate,
Università del Piemonte Orientale 'Amedeo Avogadro', corso T.Borsalino 54,
15100 Alessandria, e-mail: [email protected]
FERRERO Enrica (c/o Boero)
FERRI Franca (c/o Bartolini)
FIORI Carla Dipartimento di Economia Politica, Università di Modena, , via
Berengario 51, I-41100 Modena, e-mail: [email protected]
184
FOGLIA Serafina, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
FRANCHINI Daniela Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
FURINGHETTI Fulvia, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, e-mail: [email protected]
GABELLINI Giorgio Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
GAGGERO Maria Teresa, Nucleo di Ricerca Didattica MaCoSa, Dipartimento
di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
GALLINO Maria Gemma, via Saorgio 89/B, 10100 Torino, Italia, e-mail:
[email protected]
GALLO Elisa, Dipartimento di Matematica, Università di Torino, Via Carlo
Alberto 10, 10123 Torino, fax 39011.6702878, e-mail: [email protected]
GALLOPIN Paola, Istituto Scolastico "A.Manzoni", 33052 Cervignano (GO)
GARUTI Rossella, via del Melograno 8, 41010, Fossoli di Carpi (MO), e-mail:
[email protected]
GAZZOLO Teresa (c/o Boero)
GHERPELLI Loredana, via Liguria 1, 41100, Montale Rangone (MO)
GRANDE Rocco, Contrada Botte 28, 85100 Potenza
GREGORI Silvano, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
GIROTTI Giuseppe Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
GRECO Simonetta, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
GRIGNANI Teresa (c/o Pesci)
GRILLO Brigida, c/o Gruppo di Ricerca Insegnamento delle Matematiche,
Dipartimento di Matematica e Applicazioni - Via Archirafi, 34 - 90123
Palermo
GRUGNETTI Lucia, Dipartimento di Matematica, Università di Parma, Strada
D'Azeglio 85/A, 43100 Parma, e-mail: [email protected]
GUALA Elda, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, e-mail: [email protected]
IACOMELLA Alba, c/o Dipartimento di Matematica dell'Università di Lecce,
Strada per Arnesano, 73100 Lecce
IADEROSA Rosa, via XXV aprile 5, 200090 Cesano Boscone (MI), e-mail:
[email protected]
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JOO Carla (c/o Pesci)
LANCIANO Nicoletta, Dipartimento di Matematica, Università La Sapienza,
Piazzale A. Moro 2, 00185 Roma, e-mail: [email protected]
LANZI Elena (c/o Pesci)
LEMUT Enrica, Istituto per la Matematica Applicata (CNR), Via De Marini, 6,
16149 Genova, e-mail: [email protected]
LETIZIA Angiola, Dipartimento di Matematica dell'Università di Lecce, Strada
per Arnesano, 73100 Lecce, e-mail: [email protected]
LOCATELLO Silvano, Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
MADDALOSSO Mirella, Nucleo di Ricerca in Didattica della Matematica, ,
Università degli Studi di Padova, Dipartimento di Matematica Pura ed
Applicata, via Belzoni 7, 35131 Padova, fax +049-8275892
MAGENES Maria Rosa (Pesci)
MALARA Nicolina Antonia, Dipartimento di Matematica Pura e Applicata
'Giuseppe Vitali', Università di Modena e Reggio Emilia, via G. Campi
213/B, 41100 Modena, e-mail: [email protected]
MANCINI Marisa Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
MARCHINI Carlo, Dipartimento di Matematica, Università di Parma, Strada
D'Azeglio 85/A, 43100 Parma, e-mail: [email protected]
MARCHISIO Savina, Nucleo di Ricerca Didattica, Dipartimento di Matematica
dell’Università, Via Carlo Alberto 10 - I 10123 Torino
MARINO Teresa, Dipartimento di Matematica e Applicazioni - Via Archirafi,
34 - 90123 Palermo, e-mail: [email protected]
MARIOTTI Maria Alessandra, Dipartimento di Matematica 'Leonida Tonelli',
Università di Pisa, via F.Buonarroti 2, 56127 Pisa, e-mail:
[email protected]
MARLIA A.R., (c/o Lanciano)
MARTINI Aurelia, Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
MASI Franca Nucleo di Ricerca in Didattica della Matematica, Dipartimento
di Matematica, Università di Bologna, piazza di Porta San Donato 5, 40126
Bologna
MATTEUCCI Augusta Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
MEDICI Daniela, Dipartimento di Matematica, Università di Parma, Strada
D'Azeglio 85/A, 43100 Parma, e-mail: [email protected]
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MELONE E., Dipartimento di Matematica - II Università di Napoli, via Renella
98, Villa Vitrone - 81100 Caserta, email [email protected]
MELONI Gianna, Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
MENGHINI Marta, Dipartimento di Matematica, Università La Sapienza,
Piazzale A. Moro 2, 00185 Roma, e-mail: [email protected]
MICHELETTI Chiara, via Cibraio 64, 10144 Torino
MOLINARI Fiorenza, Unità locale di Ricerca Didattica, Dipartimento di
Matematica, via D'Azeglio 85, 43100 Parma
MORELLI Aldo, Dipartimento di Matematica e Applicazioni, complesso
Universitario "Monte S. Angelo", via Cinzia, 80126, Napoli; tel. 081675614,
fax 0817662106, e-mail: [email protected]
MORTOLA Carlo, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
NAVARRA Giancarlo, via Cugnach 4, 32036 Sedico (Belluno), e-mail
[email protected]
OLIVERO Federica, Graduate School of Education, University of Bristol, 35
Berkeley Square, Bristol BS8 1JA, UK, e-mail: [email protected]
OLIVIERI Giovanni (c/o Accascina)
PAOLA Domingo, via Canata 2/31, 17021 Alassio (SV), e-mail:
[email protected]
PARENTI Laura, Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, e-mail: [email protected]
PARODI G.P. (c/o Accascina)
PASCUCCI Nicoletta Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
PEDEMONTE Bettina, via Custo 4/12 - 16162 Genova, e-mail: [email protected]
PELLEGRINO Consolato, Dipartimento di Matematica Pura e Applicata
'Giuseppe Vitali', Università di Modena e Reggio Emilia, via G. Campi
213/B, 41100 Modena, e-mail: [email protected]
PERELLI D’ARGENZIO Maria Pia (c/o Bagni)
PESCI Angela, Dipartimento di Matematica – Università, Via Abbiategrasso
215, 27100 Pavia, e-mail: [email protected]
PIEROTTI A. (c/o Lanciano)
PINCELLA Maria Grazia , via Kennedy 41, 46047 Porto Mantovano (Mantova)
POLI Paola, IRCCS Stella Maris, viale del Tirreno 347, Calambrone (PI)
POLO Maria, C.R.S.E.M. (Centro di ricerca e sperimentazione dell'educazione
matematica), Dipartimento di Matematica, viale Merello 92, 09123 Cagliari,
e-mail: [email protected]
187
PUXEDDU M.R., c/o C.R.S.E.M. (Centro di ricerca e sperimentazione
dell'educazione matematica), Dipartimento di Matematica, viale Merello 92,
09123 Cagliari
PUXEDDU S., c/o C.R.S.E.M. (Centro di ricerca e sperimentazione
dell'educazione matematica), Dipartimento di Matematica, viale Merello 92,
09123 Cagliari
REGGIANI Maria, Dipartimento di Matematica – Università, Via
Abbiategrasso 215, 27100 Pavia, [email protected]
RIGATTI LUCHINI Silio (c/o Bagni)
RINALDI Maria Gabriella, Dipartimento di Matematica, Università di Parma,
Strada D'Azeglio 85/A, 43100 Parma, [email protected]
RIZZA Angela, Unità locale di Ricerca Didattica, Dipartimento di Matematica,
via D'Azeglio 85, 43100 Parma
ROBOTTI Elisabetta, via D.Ghelfi, 6/3 - 16164 Genova
ROBUTTI Ornella, Dipartimento di Matematica, Università di Torino, 10123
Torino, e-mail: [email protected]
ROCCO Marina, Scuola Media Statale Divisione Julia, Viale XX Settembre,
34100 Trieste
ROHR Ferruccio (c/o Accascina)
RUSSO Elvira, Dipartimento di Matematica e Applicazioni “R.Caccioppoli”,
Università “Federico II”, via Cintia, Complesso Monte S. Angelo - 80126
Napoli, email [email protected]
SACCO Maria Piera, Nucleo di Ricerca Didattica, Dipartimento di
Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino
SALTARELLI L. (c/o Cannizzaro)
SANDRI Patrizia Nucleo di Ricerca in Didattica della Matematica,
Dipartimento di Matematica, Università di Bologna, piazza di Porta San
Donato 5, 40126 Bologna
SASSI Elena, Dipartimento di Scienze Fisiche, Università "Federico II", Via
Cintia, Complesso Monte S. Angelo - 80126 Napoli, email: [email protected]
SCALI Ezio (c/o Boero)
SCIMEMI Benedetto, Dipartimento di Matematica Pura e Applicata, via
Belzoni 7, 35131 Padova, e-mail: [email protected]
SIBILLA Alfonsina (c/o Boero)
SOMAGLIA Annamaria, via Shelley 134, 16148 Genova
SORZIO Paolo, Dipartimento di Educazione, Università di Trieste, Via Tigor
22, 34127 Trieste, fax +040-6763620, e-mail: [email protected]
SPAGNOLO Filippo, Dipartimento di Matematica e Applicazioni, Università di
Palermo,
Via
Archirafi,
34,
90123
Palermo,
e-mail:
[email protected]
188
TANZI CATTABIANCHI Maria, Nucleo di Ricerca Didattica , Dipartimento di
Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino
TAZZIOLI Rossana, Dipartimento di Matematica, Città Universitaria, viale
A.Doria 6, 95125 Catania, e-mail: [email protected]
TESTA Claudia, Nucleo di Ricerca Didattica , Dipartimento di Matematica,
Università di Torino, Via Carlo Alberto 10 , 10123 Torino
TIRAGALLO Gabriella, Nucleo di Ricerca Didattica MaCoSa, Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova
TIZZANI Paola (c/O BOERO)
TOMASSINI Francesca, Gruppo di Ricerca Didattica c/o Dipartimento di
Matematica Università, Via Abbiategrasso 215, 27100 Pavia
TONELLI M. (c/o Zan)
TORTORA Roberto, Dipartimento di Matematica e Applicazioni, complesso
Universitario "Monte S. Angelo", via Cinzia, 80126, Napoli; tel. 081675614,
fax 0817662106, e-mail: [email protected]
USELLI Elsa, c/o C.R.S.E.M. (Centro di ricerca e sperimentazione
dell'educazione matematica), Dipartimento di Matematica, viale Merello 92,
09123 Cagliari
VACCARO Virginia, Dipartimento di Matematica e Applicazioni
“R.Caccioppoli”, Università “Federico II”, Via Cintia, Complesso Monte S.
Angelo - 80126 Napoli, email: [email protected]
VENÈ Margherita, Dipartimento di Matematica, Università di Parma, Strada
D'Azeglio 85/A, 43100 Parma, e-mail: [email protected]
VERCESI Nicoletta, Gruppo di Ricerca Didattica c/o Dipartimento di
Matematica Università, Via Abbiategrasso 215, 27100 Pavia
VIGHI Paola, Dipartimento di Matematica, Università di Parma, Strada
D'Azeglio 85/A, 43100 Parma, e-mail: [email protected]
ZAN Rosetta, Dipartimento di Matematica 'Leonida Tonelli', Università di
Pisa, via F.Buonarroti 2, 56127 Pisa, e-mail: [email protected]
ZUCCHERI Luciana, Dipartimento di Scienze Matematiche, Università di
Trieste, piazzale Europa 1, 34127 Trieste, e-mail: [email protected]
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