Alcuni abstract delle comunicazioni (some abstracts):
ELEMENTARY GEOMETRICAL MODELLING OF SUNSHADOWS: SOME CONDITIONS AND CONSEQUENCES
Paolo Boero, Dipartimento di Matematica dell'Università di Genova e IMA del CNR
Ezio Scali, Nicoletta Sibona, Scuola Elementare di Piossasco e IMA del CNR
ABSTRACT
Our contribution, based on teaching experiments performed in different
school and socio-cultural environments in Hungary, Italy and Spain and some
preceding published studies about them (cfr. Boero & al, Proc. PME-XIX;
Dettori & al, Proc. PME-XX; Scali, Proc. CIEAEM-46 and CIEAEM-49) will
concern:
- the conditions under which elementary geometrical modelling of sunshadows
can take place in primary school classes (grades III and IV): we will
consider students' cultural background (including a discussion about
cultural obstacles) and teacher's management of classroom activities;
- the consequences of elementary geometrical modelling of sunshadows: we
will consider changes in the verbal description of the phenomenon and their
possible interpretations; changes in the conception of the phenomenon;
development of geometrical skills and concepts.
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Analogical and digital artifatcts: cultural and didactical problems
Carlo Dapueto - Dipartimento di Matematica - Università di Genova
Abstract
Through some examples, a short survey of relations between digital and analogue, discrete and continuous, numerical and graphical, ... is presented and some didactical problems are posed:
- the overlap of algorithmic and analogic mental procedures in the first uses of graphical representations of numbers,
- the vicious circles which often are present in the use of graphic representations to justify numerical properties,
- the question of identifying the nature of the communicative power of graphic representation,
- the overlap of normative aspects (mathematics used for building the object) and descriptive aspects (mathematics for describing the object) present in activities which link numerical and graphic aspect (specially when using computer) and the opportunities for new ways of presenting (or giving motivation to studying) some mathematical concepts which this overlap offers.
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The language of mathematics from the standpoint of pragmatics
Pierluigi Ferrari - Dipartimento di scienze e tecnologie avanzate - Università del Piemonte Orientale
Abstract
A number of studies have dealt with the role of language in advanced
mathematical thinking. Some of them have analysed the language of
mathematics from the standpoint of semantics. In this paper I argue that
some behaviors may arise from the application to mathematical language of
some conventions of ordinary language; at this regard, an analysis based on
some ideas from pragmatics seems appropriate. I use Grice's Cooperative
Principle (CP) in order to carry out an exploratory investigation on some
episodes that are not easily accounted for in terms of semantics only. Some
examples of (undue) application of CP to mathematical language are given. I
argue that the application of CP to mathematical language in problem
solving is closely linked to the poor use of mathematical knowledge and,
more generally, to the so-called 'pseudo-analytical' behaviors.
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Rappresentazioni (iconiche,
proposizionali) e presentazioni grafiche: aspetti
cognitivi
Alberto Greco - Dipartimento di Scienze Antropologiche - Università di Genova
Abstract
This is a preliminary report about some aspects of the
analogical-digital representation in mathematics learning, from a cognitive
science perspective. The following issues will be addressed:
- differences between iconic and propositional representation,
and problem of their integration;
- differences and relationship between iconic and mental
representations
and graphic presentations;
-how graphic presentations support representations (such as
mathematical situations and problems; mathematical and cognitive operations)
that are relevant in mathematical teaching.
Lo scopo della relazione è di fornire un chiarimento
preliminare su alcuni aspetti della distinzione fra rappresentazioni analogiche
e digitali nell'apprendimento della matematica, che si pongono dal punto
di vista della scienza cognitiva. In particolare, vengono trattati i seguenti
punti:
- differenze tra rappresentazioni iconiche e proposizionali
e problema della loro integrazione;
- differenze e relazioni tra rappresentazioni mentali
iconiche e presentazioni grafiche;
- in che modo le presentazioni grafiche facilitano le
rappresentazioni rilevanti nella didattica della matematica: situazioni matematiche, problemi matematici, operazioni (cognitive e matematiche).
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From Plane Representation of Space Situations
to "Geometry" of Representation: a teaching experiment in the 6-th and 7-th levels
Laura Parenti - Dipartimento di MAtematica - Università di Genova
Abstract
In this work we examine the teaching experiment that enabled us to provide 11-12 year-old-children with elements of critical analysis and theorical interpretation (elements of geometry of central projection) of plane representation of space situations.
These activities involved about 80 pupils, who follow the Genoa Project, for about 15 hours in the sixth level, 25 hours in the seventh level. About one third of the pupils were school-integrated, interested, but with scarce results in learning (in mathematics as well as in the other disciplines).
We suggested meaningful problem-solving geometric situations that arise as problems of plane representation of space situations, require to produce conjectures, allow mental experiments based on dynamical explorations and find a theorical frame in elements of geometry of central projection the class manage to construct and must master.
We will show particularly how, according to our hypotheses, pupils can model objects (which are represented or must be represented) not only according to their figural aspect (perspective elements), but also according to their conceptual aspect if, with the teacher's help, they succeed in finding out a system of relationships which "rules" the geometrical transformations inherent to geometry of the representation. The theorical model that is found in this way (elements of geometry of central projection) can also become an instrument to validate situations which cannot be decided in an empiric way (that is only by physical experience) and to foresee outcomes that can be noticed in reality, through an early approach to theorems and related processes.
In the work we will also show how pupils' awareness of their own mental dynamics and their practice to manage them can have a good influence on the production of conjectures and proofs.
We will notice as well how the mental processes, which are involved in the proof (meant as a logic deductive process within the theory of reference) can be influenced by the dynamic approach.
We are not going to analyse the teacher's role in details, but we want to mention it is very important both in class as a cultural mediator, who coordinates and enhances discussions, and in the research group, in order to plan activities and analyze the pupils' mental processes.
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