angela pesci

Sommario Si descrivono i risultati principali ottenuti da analisi di esperienze di insegnamento della matematica nella scuola secondaria, svolte in presenza con l'attuazione di due modelli di cooperazione fra pari: la modalità collaborativa e il tutoraggio. Le due modalità si differenziano per i ruoli previsti (cinque nel primo caso e due nel secondo) e per la tipologia di compiti da assegnare agli studenti. Dalle analisi svolte è emerso che il modello collaborativo è più adatto ad affrontare compiti nuovi o abbastanza complessi, il tutoraggio è più efficace nel recupero di competenze matematiche. Entrambi i modelli sono risultati particolarmente efficaci per promuovere, attraverso il dialogo fra pari, compiti di natura metacognitiva. L'analisi presentata può essere significativa per la rivisitazione delle modalità collaborative descritte in vista di una loro implementazione online. PAROLE CHIAVE Collaborazione fra pari, Tutoraggio fra pari. Abstract This contribution descri...

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 resolutive 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. 1.

After the reprise of a model of intervention for the training of mathematics teachers (both initial and in-service) developed after experiences carried out in a cooperative modality (Pesci, 2007a), several tasks are presented for encouraging the development of disciplinary, didactic, and relational competences of the teachers. The theoretical framework related to these tasks puts in evidence the reasons of their choice: the importance, for teachers, of collaboration in sharing personal experiences, difficulties, and resource, the importance of autobiographical reflection, of reflection on one's own classroom practices, and of epistemological reflection on the disciplinary contents. The connection to the debate about tasks which is developing considerably in relation to the education of teachers (Javorski, 2007) is underlined.

This contribution describes the main results from analysis of face-to-face experiences in secondary school mathematics classes regarding two models of peer cooperation: the collaborative model and peer tutoring. The models differ in terms of the roles required (five in the former case, two in the latter) and in the tasks assigned to students. The analysis indicates that the collaborative model was more suitable for tackling new and complex tasks, while peer tutoring proved more effective for remediation of mathematics skills. Both models were highly effective for promoting peer dialogue and metacognitive reflection. The reported analysis may prove useful for re-elaborating the two collaborative models for the purposes of online implementation.

To investigate students’ ways of working with concrete materials in mathematics, a three-dimensional static artefact was constructed and made available to upper secondary students, with pre-knowledge only in two-dimensional coordinate geometry, for solving problems about planes and straight lines in space. Artefact interactivity was generally high, even students also disregarded the model to work only numerically with the coordinates, building on knowledge about lines in two dimensions. The model was used when trying to convince other students in the group.

After an introduction on the importance of redefining the didactic system in the terms of a more global interpretation of the personal relationships which intertwine in it, we look at the general principals of “cooperative learning”, an educational strategy by social mediation in which the resources for the construction of knowledge are the students, who are called upon both to accomplish a disciplinary task and to develop social abilities. The role of the teacher remains fundamental, being the organiser and facilitator of the entire process. After the presentation of some didactic experiences carried out in the upper secondary school with the cooperative groups, some elements of the evaluation of the experiences carried out are put forth, also through the judgments expressed by the students themselves. At the end, note is made of some open problems which it seems necessary to confront.

This paper is divided into three main sections. In the first one, Sect. 2, there is a collection of contributions from recent literature about the processes implied by terms such as to visualize, to imagine, etc. reported by researchers in mathematics education. It is clear that the objects of such processes are mental images, but the debate about the nature of mental images, though interesting, will not be considered here. Some statements on the role of graphical representations in mathematics activities will also be reported.

Starting from the first half of the last century, with the fundamental contribution of the studies of Piaget, the conviction that the meaningful construction of knowledge can only come about with the active contribution of the learner was developed and largely shared. The educative process must be adequately ignited by the teacher, but the real protagonist of the entire process turns out to be the pupil; responsible in the first person for the construction of his own ideas. The constructivist ideas of Piaget, taken up and developed with different emphases by numerous scholars (one thinks for example of the radical constructivism of V. Glasersfeld (1995) or of the social constructivism of P. Ernest (1995) or of the theory of didactic situations of G. Brousseau (1997)) are well known today, having been amply spread through articles on didactic research. Also, in the Italian programmes of 1990 for the upper secondary school, in the “ General Didactic Directions” it says, for example, t...

After an introduction on the importance of redefining the didactic system in the terms of a more global interpretation of the personal relationships which intertwine in it, we look at the general principals of “cooperative learning”, an educational strategy by social mediation in which the resources for the construction of knowledge are the students, who are called upon both to accomplish a disciplinary task and to develop social abilities. The role of the teacher remains fundamental, being the organiser and facilitator of the entire process. After the presentation of some didactic experiences carried out in the upper secondary school with the cooperative groups, some elements of the evaluation of the experiences carried out are put forth, also through the judgments expressed by the students themselves. At the end, note is made of some open problems which it seems necessary to confront. 1.